Nonlinearity & Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis

NONLINEARITY AND FUNCTIONAL ANALYSIS Lectures on Nonlinear Problems in Mathematical Analysis This is a volume in PURE...

Author: Melvyn S. Berger

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NONLINEARITY AND FUNCTIONAL ANALYSIS Lectures on Nonlinear Problems in Mathematical Analysis

This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks

Editors: SAMUEL EILENBERC AND HYMAN BASS A list of recent titles in this series appears at the end of this volume.

N0NLINEARITY AND FUNCTIONAL ANALYSIS Lectures on Nonlinear Problems in Mathematical Analysis

Melvin S. Berger Belfer Graduate School Yeshiva University New York, New York

ACADEMIC PRESS

W New York San Francisco

London 1977

A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT 0 1977, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART O F THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED I N ANY F OR M OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United KitiPdom Edition oublished bv ACADEM~CPRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1

Library of Congress Cataloging in Publication Data Berger, Melvyn, Date Nonlinearity and functional analysis. (Pure and applied mathematics) Bibliography: p. Includes index. 1. Mathematical analysis. I. Title. theories. QA300.B45 8 515'.7 ISBN 0-12-090350-4

2.

Nonlinear 76-26 0 39

PRINTED IN THE UNITED STATES O F AMERICA

Section 6.2A, pp. 313-317, is reprinted with permission of the publisher, American Mathematical Society, from the Proceedings of SYMPOSIA IN PURE MATHEMATICS, Copyright @ 1970, Volume XVIII, Part I, pp. 22-24.

To the memory of my father, Abraham Berger

How manifold are Thy works, 0 Lord! In wisdom hast Thou made them all. PSALMS 104

The approach to a more profound knowledge of the basic principles of physics is tied up with the most intricate mathematical methods. A. EINSTEIN

All depends, then, on finding out these easier problems, and on solving them by means of devices as perfect as possible and of concepts capable of generalization. D. HILBERT

CONTENTS

xiii xvii xix

Preface Noiation and Terminology Suggestions f o r ihe Reader

PART I Chapter 1

PRELIMINARIES Background Material

I . 1 How Nonlinear Problems Arise 1. IA Differential-geometric sources l . l B Sources in mathematical physics I Classical mathematical physics I 1 Contemporary mathematical physics I.1C Sources from the calculus of variations

3 3 9 10

14 16 18

1.2 Typical Difficulties Encountered 1.2A Inherent difficulties 1.28 Nonintrinsic difficulties

18 21

1.3 Facts 1.3A I .3B 1.3C 1.3D 1.3E 1.3F

25 25 26 30 31 32 35

from Functional Analysis Banach and Hilbert spaces Some useful Banach spaces Bounded linear functionals and weak convergence Compactness Bounded linear operators Special classes of bounded linear operators

1.4 Inequalities and Estimates 1.4A The spaces W , , ( Q ) (1 I p a) 1.4B The spaces W,",,(RN) and W,,,,,(Q) ( m 2 1, an integer, and Isp 2.

(v) Mapping properties of hoiomorphic functions Nonlinear partial differential equations analogous to (1.1.5) arise quite naturally in the geometric study of holomorphic functions of one complex variable. Let f be a holomorphic mapping of the unit disk D equipped with the Poincare metric ds2 = (1 - z,T)-~ dz dZ into the extended complex plane. Let the metric dS2= df d j be defined on f ( D ) , and set e" = (ds/dS)*. Then the Laplace-Beltrami operator A relative to dS2 can be written A M = - +(a2u/af and a short computation shows that u satisfies the equation AM = 2e". This equation is independent off and was used by Poincare in his studies on automorphic functions and later by F. Nevanlinna in his differential geometric proof of the value distribution of meromorphic functions. (vl) Deformations of complex structures Deformations of the complex structure on a compact manifold Em' of complex dimension 1 were first studied by Riemann in 1857. He found that the number m(3n')of independent complex parameters on which the deformation depends can be completely described in terms of the Euler characteristic of Gx' (or equivalently, in terms of the genus of the Riemann surface associated with %'). fiemann called m(Gx') the number of moduli, and the study of

HOW NONLINEAR PROBLEMS ARISE

1.1

9

these complex parameters has occupied the attention of a great many researchers down to the present day. Thus two Riemann surfaces may be topologically equivalent, but the “moduli” of these surfaces determine their analytic equivalence. For higher dimensional complex manifolds W , the analogous deformation problem is less well understood, and (in contrast to the onedimensional case) has a high& nonlinear character. To illustrate this point, let Em” be given a complex structure_ V,, with suitable distinguished complex coordinates z,, . . . , z,. Let V be another complex structure underlying Em” with local coordinates y , , . . . ,y,, and such that the (1, 0) forms can be written dyj = dzj

+

‘pkj &k

with ‘pkj small.

k

Then f is said to be an almost complex structure near the complex if and structure V,,. Then f will define a true complex structure on an only if the vector-valued (0, 1) form w = WF dZk

satisfies an appropriate integrability condition. This condition takes the form of the nonlinear partial differential equation (1.1.7)

?w = [a,w ] .

Here the linear differential operator 8 is the canonical operator that maps vector-valued (0, 1) forms into vector-valued (0,2) forms by the rule

and the bracket [w, 01 is a certain bilinear vector-valued (0, 2) form. Consequently, to study the complex structures on W near V,,, we need only study the solutions w of (1.1.7) that lie sufficiently close to zero. This notion distinguishes a nonlinear aspect of the deformation problem if n 2 since for n = 1 almost all complex structures are automatically integrable, the bracket [a,w ] vanishes identically and consequently the system (1.1.7) is linear. For a further discussion, the reader is referred to Chapter 4 and the references quoted there. As we shall see, the resolution of this deformation problem is greatly simplified by the introduction of Hilbert space techniques together with notions of “bifurcation” theory. 1.1B

Sources in mathematical physics

Another equally rich source of nonlinear differential systems can be found in the basic problems of mathematical physics. The following general examples are among those discussed in the sequel.

10

I

BACKGROUND MATERIAL

I Classical mathematical physics

(I) Newtonian mechanics of partlcles Consider a system of N particles pi of mass mi ( i = 1, . . . , N ) moving in R3 subject to forces derived from a potential function U ( x , , . . . , x j N ) .The motions of these particles are found as solutions of the differential system

This system is clearly nonlinear when U is not a quadratic function of its arguments, but depends on its arguments to some higher order. Now a fundamental problem for classical mechanics is the determination of the periodic motions of (1.1.9) for various reasonable potential functions U.The importance of periodic motions resides in their observation for many diverse natural phenomena governed by equations of the form (1.1.9). Moreover, Poincare has conjectured that the periodic solutions of (1.1.9) (for appropriately restricted U)are “dense” in the set of all solutions. Here, density means given any solution x ( t ) , then there is a periodic solution differing only slightly from x ( t ) for a given length of time. When the forces acting on the particles are of a purely gravitational nature, Newton’s law of gravitation implies

and the resulting system (1.1.9) represents the governing equations for the classically formidable N-body problem. In celestial mechanics, the wellknown two-body or Kepler problem can be solved quite explicitly and is consequently important since many problems of astronomy can be regarded as perturbations of it. Indeed, one such perturbation, the wellknown restricted three-body problem, was considered by Poincare to be typical of general dynamical systems. As an even simpler example, the equations of motion governing an autonomous perturbation cf(x) of the Kepler problem can be written (1.1.10)

x

X +3

+€f(X) = 0, xERN 1x1 At x = 0, the term x/lxl’ has a singularity. The removal of the difficulties

inherent in this fact has given rise to a rather elaborate “regularization” theory, in which one avoids analysis of (1.1.10) near x = 0 by appropriate changes of coordinates in (1.1.10) on a fixed energy surface (cf. Chapter 6). The system (1.1.10) is then reduced to the form y2 + W(y) = const. y + grad W(y) = 0, where W(y) is a smooth function vanishing at y = 0 to the second order.

1.1


11

Classical methods for studying the periodic solutions of such nonlinear systems as ( I . 1.9) often break down due to “resonance” effects (among others). This fact has given rise to many new attempts to utilize topological methods in studying such problems, and we shall discuss this topic in Chapters 4 and 6. (11) Elasticity A deformable body B is called elastic if it may be deformed by the application of a given class of forces to part of B , but returns to its original state after the forces are withdrawn. The simplest classical formulation of elasticity is based on the two assumptions of a linear stress-strain law (Hooke’s law) and of infinitesimally small displacements. These assumptions imply linear governing equations. However, if one takes into account possible large deformations produced but retains Hooke’s law, the resulting equations governing the equilibrium states of the elastic body B are nonlinear. Thus, the equations governing the equilibrium states of a one-dimensional elastic body B (a rod) under the action of compressive forces of magnitude h applied to its ends can be written as the boundary value problem 1/2

-

(1.1.11) w,,+Aw(l-w,2) -0, w(O)=w(l)=O. Here w is a measure of the horizontal deflection produced in B by the compressive force. This classic system is known as the elastica problem of Euler since it was solved completely by him in 1744. Its two-dimensional analogue (discussed by von Karman in 1910) concerns the deformations produced in a two-dimensional elastic body B of arbitrary shape c R2 (a thin elastic plate) acted on by compressive forces of magnitude h on its boundary. This problem is considerably more difficult than the onedimensional case. By utilizing the modern techniques described in the sequel, research has only recently begun to give an adequate mathematical treatment of the problem. The resulting deformations are governed by a system of two coupled partial differential equations, the so-called von Karman equations, defined on a, and can be written (after suppressing certain physical parameters) ( 1.1.12) A2F= - + [ w , w ] , e’Ab=[f,w],

where A2 denotes the biharmonic operator and

t

f?

81 = fx&y

+ &XX

- 2fxygxy’

Here c 2 is a measure of the thickness of the thin plate, w represents the vertical deflection of B from its undeformed state, and F represents the Airy stress function from which all the stress components of the deforma-

12

I

BACKGROUND MATERIAL

tion can be found. Although the deformations predicted from (1.1.1 1) can be found explicitly in terms of elliptic functions, the integration of (1,1.12), in general, can be understood only by a careful qualitative analysis using the methods to be described in the sequel. The equations (1.1.12) possess many subtle properties, and we shall use these in the sequel as a specific, nontrivial example of our theoretical developments. (iii) Ideal incompressible fluids The velocity distribution u of an ideal incompressible fluid is governed by the (nonlinear) Euler equations of motion and the equation of continuity. Denoting the velocity components by ui, the density of the fluid by p, and the pressure by p , and assuming the fluid is acted on by forces Fi,these equations are

(1.1.14)

div u = 0.

Assuming that (i) the flow is irrotational, so that the velocity vector is the gradient of a velocity potential 5 ; (ii) gravity is the only force acting on the fluid; (iii) the flow is stationary; and (iv) the flow is two dimensional; the system (1.1.13) has the first integral (1.1.15)

+(l:,

+ 5.:2) + gx,

= const.

on

ar,

and (1.1.14) becomes (1.1.16)

Ag=O

on

r.

The nonlinear aspect of this problem is twofold. The boundary ar of r is unknown, and the boundary condition imposed on ar is nonlinear. Solutions of the system (1.1.15)-( 1.1.16) have great importance in the theory of water waves, a subject renowned for its many interesting local and global nonlinear phenomena (see Section 5.5). The vortex motions of an ideal incompressible fluid (whose study was initiated by Helmholtz in 1858) exhibit particular striking nonlinear phenomena. Consider, for example, vortex rings of a permanent form that can be observed in such a fluid. By a vortex ring we mean a continuous axisymmetric solenoidal vector field q defined on R 3 and a subset Z of R 3 (homeomorphic to a solid torus) such that (taking axes fixed in Z), both q and 2 do not vary with time, the vorticity w = curl q vanishes outside Z (but not in 2 ) and moreover satisfies the Euler equations of motion (1.1.13) and the appropriate boundary condition at infinity. In the sequel (Section 6.4) we shall derive and study the following semilinear elliptic partial

13


1.1

differential equation for the “Stokes stream function” I,L associated with q:

Here the given function f governs the distribution of vorticity in 2, while both 1c, and its gradient must be continuous across the boundary of 2, d 2 . Again, as in (1,l. 15)-( 1.1.16), the equation (1.1.17) is nonlinear in two respects: both \i/ and 2 must be determined from it and the appropriate boundary conditions. Classically, two extreme explicit solutions of (1.1.17) were known. The problem of finding a one-parameter family of vortex rings joining these extremes requires global methods and will be discussed in Chapter 6. (See Fig. 1.1 .)

FIG. 1.1 Vortex ring distributions in R3 illustrating the intermediate solid toruslike vortex rings of varying cross sections interpolating between the classical Helmholtz singular vortex ring and Hill’s spherical vortex.

(iv) Viscous incompressible fluids The equations governing the velocity distribution of a viscous incompressible fluid, the so-called NavierStokes equations, are (1.1.18)

all, +

-

at

(1.1.19)

,

du,

I,

~

1

a,yj

=

-

1 P

-

aP ~

ax,

+Y

Au, + F,

( i = 1. 2, 3 ) ,

div u = 0,

and so differ from the Euler equations only by the addition of the terms

14

I

BACKGROUND MATERIAL

v Aui. Here v is a measure of the viscosity of the fluid. If the fluid under consideration occupies a region C2 with boundary i3 C2, one generally adds a homogeneous or inhomogeneous boundary condition uiJan = gi, which implies in the homogeneous case that the velocity of u at aC2 vanishes. These equations describe a vast range of observed hydrodynamic behavior for both large and small values of the viscosity v. I t is still an open problem to show that the complicated phenomena of turbulence can be described on the basis of the nonlinearity of these equations, although as we shall see in Chapter 4, the onset of turbulence, via the appearance of secondary solutions, can be rigorously established in many instances, as a bifurcation phenomenon. II Contemporary mathematical physics (I) Theory of quantum flelds The principle of superposition definitely rules out the appearance of nonlinear equations governing elementary quantum-mechanical phenomena. However, as soon as one considers the interaction of various quantum fields, the possibility of nonlinear equations of motion reappears. In fact, in recent years, some success has been achieved by the study of the following Lorentz-invariant nonlinear Klein-Gordan equation

+

(1.1.20) lt,- A{ = -m2{ F(l{\’){ and its generalizations. Here { is a complex-valued “wave” function. An interesting nonlinear problem in quantum field theory, posed by A. Wightman, concerns the notions of dynamical instability and broken symmetry for model theories. In a simple (mean field) approximation (and rewritten in terms of contemporary “Euclidean field theory”) such questions can be studied relative to the equation

Au - m2u + P ’ ( u ) = f m (defined on RN), where f m is regarded as a constant given a priori, and P’(u) is a polynomial-like function of u such that p ( ~ ) +- 00 as IuI + co. The mathematical problem to be studied concerns the minima of the associated functional

(1.1.21)

and the question of its uniqueness and existence relative to various choices of the constantf, and the zeros of the function - m2u + P’(u). The idea is that by perturbing the right-hand side of (1.1.2 1) by various perturbations of f m , different unique absolute minima of (1.1.22) result, and these may help account for the appearance of the various so-called strange particles

1.1


15

occurring in contemporary field theory. (For more details see Section 6.2.) See Fig. 1.2 for a pictorial view of this approach to dynamic instability.

\

IfynrmlL \ l r h l l l t y

FIG. 1.2 Qualitative behavior of dynamic stability versus dynamic instability in quantum field theory models.

(11) Relativistic theory of gravitation According to the general theory of relativity, space-time is represented by a four-dimensional (normal hyperbolic) manifold ( V4,g ) that has an indefinite metric ds2 = gap dx" d x B(a, p = 1, 2, 3, 4). Gravitational effects on test particles and light rays are described by saying that the motions of such entities are geodesics with respect to the metric g . The ten components gap of the metric g are not arbitrary in Einstein's theory, but must satisfy certain nonlinear partial differential equations. For free space, these equations can be written as Rap = 0, (1.1.23)

where R,, is the Ricci tensor with respect to the metric gap. A timeindependent, radially symmetric solution of this quasilinear system can be found, which, in addition, has the property that it is asymptotic at large distances to the Lorentz metric ds2= dt2 - dx: - dx: - dxf. This solution, the so-called Schwarzschild metric, can be written in spherical polar coordinates as (1.1.24)

ds2=

dr

-r2{df12+sin2fldq2).

16

I

BACKGROUND MATERIAL

However, in general, the nonlinearity of the system (1.1.23) and its generalizations has caused great difficulties in exploring the further implications of Einstein's theory. (Ill) Phase transitions in solids One of the most intriguing nonlinear problems in contemporary mathematical physics is the theory of phase transitions. From the point of view of thermodynamics, the sharp transition of a substance from one state to another can be understood in terms of the Gibbs internal energy function U ( x l , . . . , x,+ I), where the variables x , stand for appropriate generalized coordinates. For a broad class of systems, the function U is homogeneous of degree 1, so it is customary to normalize U by setting x , + ~= 1 and scaling U accordingly. Moreover, a fundamental physical postulate of thermodynamics states that the stable equilibrium state of the system described by U is the minimum of U. Now the stability of the system at a given equilibrium state X is expressed by the if this form is indefinite, definiteness of the quadratic form X I , ,UX,,(X)5,5,; then the state 7 cannot exist in a homogeneous form but breaks up into two or more phases, each of which satisfies the stability condition. In applications, states with semidefinite forms are of special interest and correspond to generalized critical points of the system. A deeper study of the phase transition problem going beyond a thermodynamic interpretation has been established by Onsager for a particular model, the twodimensional Ising model. This was accomplished by explicit computation, and it is an important unresolved problem to study such phase transitions sdccessfully by qualitative methods, especially in three dimensions.

Sources from the calculus of variations

1.1C

A third source of nonlinear differential systems is intimately connected with the formal development of the calculus of variations. Indeed, the characterization of physical and geometric entities by extremal principles is a basic goal of a large portion of scientific thought and, for example, has survived the transition from classical to modern physics. In mathematical terms, if u ( x ) is a stationary value of some functional defined over a domain 8,

g,(w)

=k(x,Dw,. . . , D%) w,

(relative to a sufficiently large class of admissible functions satisfies the Euler-Lagrange differential equation (1.1.25)

e),then u ( x )

(-l)'"'D*F,(x,w, Dw,. . . , D " ' w ) = O .

gd(w)la1

<m

Here we have used the notation aF/aw,

=

Fa. If the dependence of F on w

1.1


17

is higher than quadratic, this differential equation will be nonlinear in w (in general). More generally, if u ( x ) is a stationary value of the functional !l0(w) subject to the integral constraints $,(w) =

Lci(x,w , . . . , D ~ W )

( i = 1, . . . , N ) ,

then u ( x ) will satisfy the equation N

(1.1.26)

2 X,-“,(w) = 0

(j=O, I..

. . ,N ) ,

j=O

where the X j are real numbers not all zero, and each g j ( w ) has the form (1.1.25). Boundary value problems associated with equations of the form ( I . 1.25) recur throughout mathematical physics. Thus it is natural to make a careful mathematical analysis of the partial differential equations defined by (1.1.25F( 1.1.26), and, in fact, we shall do so in the sequel. As well as satisfying the Euler-Lagrange equations (1.1.25), an extremal u ( x ) of 4,(w) also may satisfy natural boundary conditions. In particular, if the admissible class does not restrict the boundary behavior of the admissible competing functions, we may derive extra restrictions on an extremal by considering general variations of boundary values. Thus for the integral

e

the “natural boundary condition” at x = 0 and x = 1 is easily determined to be a F ( x , w ( x ) , w ’ ( x ) ) / a y ’= 0. Moreover the following questions arise in this connection: Does the fact that a nonlinear system is derivable from a variational principle affect the nature of its solutions? Can this variational principle be utilized in a qualitative study of these solutions? As we shall see in the sequel, the answer to these questions is a resounding yes for many of the classical problems of geometry and physics just described. Moreover, this affirmative answer also holds for general classes of nonlinear systems. The elaboration of these facts is one of the key motifs of this monograph. It is remarkable that this idea has not been utilized in most classical approaches to nonlinear problems, despite the widespread formal use of variational principles in deriving the governing equations of nonlinear systems. Furthermore, the solutions of many nonlinear problems arising as the stationary points of some functional 4 (w)are saddle points of this functional. Thus classical methods of the calculus of variations become inapplicable since these deal primarily with absolute minima. The study of vortex rings of permanent form and periodic motions of Hamiltonian

18

I

BACKGROUND MATERIAL

systems illustrates this situation. One method of treating this situation, used in the sequel, is the conversion of the problem into an isoperimetric variational problem by a clever choice of a constraint C. (See Fig. 1.3.)

-m

FIG. 1.3 Graphs of a typical energy functional before and after the imposition of a constraint C, illustrating the conversion of a saddle point to an absolute minimum.

1.2 Typical Dlfflcultles Encountered

Nonlinear systems of the type considered in Section 1.1 possess certain general properties and peculiarities that make their study both difficult and interesting. Actually, one can distinguish two classes of difficulties, one inherent in the nonlinear system itself, the other related to the manner in which such systems are generally studied. We illustrate these below by citing simple examples. 1.2A

Inherent difficulties

(I) Nonunlqueness This property is perhaps the most characteristic feature of nonlinear systems. For example, the boundary value problem for the system (1.2.1) Yff + 2 y 3 = 0,

(1.2.2) y(0) = y ( A ) = 0 has a countably infinite number of distinct solutions y N ( f )such that SUpIo,Al I Y N ( t ) l + 00 as N + 00. Indeed, solutions y ( t ) of this system passing through (0, 0) must satisfy t = /Z;(c4 - y4)-'/* dy, so that y ( t ) is a periodic function (varying between the limits a c ) and of period T = 2c-'Jil(1 - t 4 ) - ' l 2 dt = (2/c)p (say). In order that y ( A ) = 0, it must

19

1.2 TYPICAL DIFFICULTIES ENCOUNTERED

be that for some integer N , T = A / N ; so that for c = ( 2 N / A ) P (N = 1, 2, . . . ), the system (1.2.1H1.2.2) possesses the desired solutions y N ( t ) .Because of this nonuniqueness, the study of the stability of solutions of nonlinear systems assumes great importance. (11) Slngularltles The solutions of many nonlinear systems may develop singularities although the systems themselves have smooth coefficients. Consider, for example, the (Navier-Stokes-like) nonlinear boundary value problemj; + y j = O,y(O)= y(A) = 0. One readily verifies by direct integration that this system possesses no solutions without singularities however small the number A is chosen.

(111) Crltlcal dependence on parameters The structure of the solutions of nonlinear systems containing explicit parameters often change radically with the parameter. Thus the system

(1.2.3)

j;++AeY=O,

(1.2.4)

y(0)= y ( l ) = 0

has (a) no (smooth) solution for A > ,8 (a certain positive real number), (b) exactly one (smooth) solution for A = p, and (c) precisely two solutions for 0 < A < P. These facts can be proved by direct integration. Indeed, for A > 0, the solutionsy(t) of (1.2.3) passing through the origin satisfy t = A-1/2Ly(c - e’)-’/’ ds,

where c = 1

+ A-52(0);

which shows that the curve y = y ( t ) increases to a maximum at y ( t J c. Since the curvey(t) is symmetric with respect to t = t , , we require that

= log

Hence, after a simple computation, one finds cosh’(& /4) = c ; and the solutions of this transcendental equation imply ( a Hc ) above. Furthermore, as first noted by Poincare, nonlinear differential systems S, (depending explicitly on a parameter A) exhibit a large variety of bifurcation phenomena. That is, there exist certain values of A, A, say, such that s, possesses at least two distinct curves of solutions yo(A) and yl(A) with yo@)+yl(A) as A +A,. As an example, consider the system (1.2.5)

j;

+ A2{y - y’} = 0,

y(O) = y( I ) = 0. Using the Jacobi elliptic function sn(5, k ) with quarter-period K ( k ) , one

20

I

BACKGROUND MATERIAL

finds that (1.2.5) can be integrated directly. Indeed, setting 1/ 2

sn(2nK(k)x, k ) ,

< 1,

0< k

n = 1, 2 , .

..,

+

with A = 2n(l + k2)'/*K(k) 2 na, one finds that for ma < A < ( m l)a, the set of solutions of (1.2.5) is 0, ty,,. . . , + y m . Also as A+ma, y m ( x )-0 uniformly on [0, 11. Consequently, the system (1.2.5) exhibits "bifurcation" phenomena at the values A2 = m2a2. In the investigation of systems with features akin to (1.2.5), Poincare introduced the notion of exchange of stability. We say that a solution y(A) of (1.2.5) is stable if it minimizes

With this definition, it follows that yo(A) = 0 is stable for 0 < A2 Q r2,but only y,(A) is stable for all A' > r2.Thus as A2 increases through r2,the, property of stability is exchanged between the families yo(A) and yl(A). (Iv) Dlmenslon versus nonlinear growth For certain differential systems S defined on a domain a, there is a striking relationship between the growth of the nonlinear terms in S , the dimension of the set 9,and the existence of singularity-free solutions of S defined on 9.As an example, we shall show that the equation

(1.2.6)

AU - u

+ IgI"U = 0

has no nontrivial smooth solutions defined on RN(N > 2) with IuI -0 as 1x1 + 0 0 , for u 2 4/(N - 2). (In Chapter 6 , we shall show that (1.2.6) has smooth solutions for 0 < u < 4/(N - 2).) The nonexistence of smooth solutions of (1.2.6) with u 2 4/(N - 2) is obtained by proving: (*)

If u ( x ) is a solution of the equation

(1.2.7) Au+f(u) = O on with 1.1 + O as 1x1 + 00, then

RN (N

> 2)

where F,,(u) = f(u). Proot: Assuming the truth of (*) for the time being, we find that if u is a smooth solution of (1.2.6) with u2 1 and f(u) = - u + IuI% F(u) = - 2 (r + 2 luI"u2,

+

then (*) implies that

21


Thus for u # 0, u < 4/(N - 2). To prove (*), we note that a solution u ( x ) of (1.2.7) is a critical point of the functional

S(4.))

= JRN

[ ;I v

Ul2

- F ( u ) l dx,

so that we must have ( d / d k ) 4 ( u ( k x ) ) l k , ,= 0. Making the change

of variables y = kx in

5 ( ~ ( x ) ) , we find

(t) On the other hand, multiplying the equation Au we find

+ f ( u ) = 0 by u ( x ) and integrating by parts,

(tt) Combining

(t)and (tt)gives (*).

(v) Decay at infinity Another special property of nonlinear systems, exemplified in (1.2.6), can be described roughly as the amplification of decay of solutions of nonlinear elliptic differential systems defined on unbounded domains. Indeed, if u ( x ) is a solution of (1.2.6), it can also be regarded as a solution of the linear system

(1.2.8)

Au- u+p(x)u=O,

IuI+O

as

IxI-+co,

where p ( x ) = I uJ‘ -+0 as 1x1+ 00. Then, using well-known results from the linear theory of elliptic partial differential equations, one finds that 1 u(x)l = O ( ~ X \ - (~p) > 0 say) as 1x1 + co. Iterating this procedure, one eventually finds that lu(x)l = O(e-YIXI)as 1x1 + 00 for some constant y > 0. (vl) Symmetric causes may not produce symmetric effects A thin circular elastic plate is clamped along its edge and subjected to an axisymmetric uniform compression (of large magnitude) there. One observes that the elastic plate deforms into a new stable nonaxisymmetric equilibrium state under this force. The deformations observed are governed by a system of nonlinear partial differential equations due to von Karman (1.1.12), and it is the nonlinearity of these equations that gives rise to this unusual effect since the associated linear theory would predict an axisymmetric equilibrium state. 1.28 Nonintrinsic difficulties

All the above properties were connected with the intrinsic behavior of solutions of nonlinear differential systems. We now consider those difficulties associated with particular methods of studying nonlinear systems. (I) inadequacy of linearization procedures The most well-known of these methods is a process called “linearization” in which higher order

22

I

BACKGROUND MATERIAL

terms in the nonlinear system are totally disregarded locally (in the immediate vicinity of the origin (say)). Such a procedure may well yield incorrect results. Indeed, consider the structure of the nontrivial periodic solutions (near the origin) of the system

+ x - y 3 = 0 , (b) ytt + y + x3 = O . The linearized system x,, + x = 0, y,, + y = 0 possesses a four-parameter family of periodic solutions, whereas the system (1.2.9) has only the trivial periodic solution x ~y = 0. To prove this last statement, suppose the (1.2.9)

(a) x,,

system (1.2.9) had a &periodic solution ( x ( t ) , y(t)). Then multiplying (1.2.9a) by y ( t ) , (1.2.9b) by x ( t ) , subtracting, and integrating by parts (over a period), we find /{[x4(t) + y4(t)] dt = 0, so that x ( t ) = y ( t ) E 0. This procedure of linearization can be somewhat generalized by the so-called conjugacy theory of mappings. For example, if one is given the first-order system of nonlinear ordinary differential equations (1.2.10) z, = A z +f(z), where If(.)\ = o(IzI), one could attempt to find a C ' (or even continuous) change of coordinates z = ( ~ ( 5such ) that locally near z = 0, the new system could be written in its linearized form 5, = A t . Clearly, if one converts (1.2.9) to the form (1.2.10) by setting x, = u, yt = w , and z = (x, y , u, w), such a transformation cannot exist. For then nontrivial periodic solutions in 5 (near the origin) would correspond to nontrivial periodic solutions in z (near the origin) and thus to nontrivial periodic solutions ( x ( t ) , y ( t ) ) for (1.2.9). (11) Small-dlvlsor problems Another property (historically known as small-divisor problems) is a consequence of proving convergence via the Cauchy majorant method. In this method, the solutions of a large class of nonlinear differential systems are constructed as formal power (or Fourier) series in a parameter p (say), and one attempts to prove the convergence of the resulting series for a large class of values of p. Thus, for example, suppose one wishes to find a 2a-periodic solution g(z) of the difference equation g(z + 2ap) - g ( 2 ) =f(z), where f(z) is 2n-periodic. If f(z) fneinz is the Fourier series of f, then the Fourier coefficients for g =En,,, gneinZare g, = fn[e2"'p - 1]-'. For rational p some of these denominators vanish, but for irrational p, )e2ai'"'- 11 can be arbitrarily small. Nonetheless, as can be shown, for almost all p (in the sense of Lebesgue measure) (1.2.11) has a unique solution g(z) which is somewhat less smooth than the function f(z).

(1.2.11)

(111) Asyrnptotlc solutlons On the other hand, in certain problems involving power series, the convergence of these formally constructed


23

=x;=,

[,p” ( N = 1, 2, . . . ) may be difficult (if not imposseries tN(p ) sible) to determine as N -+ 60. Yet these series may be “asymptotic” to a given solution [ ( p ) of the nonlinear system, for p small, in the sense that p-N1l[( p ) - tN(p)II + O as p+O, with N fixed. Thus, for example, consider the system p2x,, - x = e - ” , x+O as Itl-+co. (1.2.12)

With q ( t ) = e - ” , the sequence x2,(t, p ) = q ( t )

+ p2q‘2’(r) +

*

*

*

+p2y-

1) q(21’4)( 2 )

is asymptotic to the unique solution of (1.2.12) for p sufficiently small, although the associated power series has radius of convergence zero. See Section 4.4. (Iv) Lack of a prlorl bounds Many methods of studying a nonlinear differential system S are based on finding bounds for all its possible solutions, depending on the special form of S . These “a priori” bounds assert the existence of certain universal constants, bounding some measure of the size of any solution u ( x ) of S. For linear systems of the form Lu = g, the nonexistence of such a priori bounds indicates that the equation Lu = g may not be solvable for all smooth functions g. However, for nonlinear systems, this fact may be quite false. Consider, for example,

(1.2.13)

yt, + y 3 = g ( 0 ,

y,(O) = y,(l)

=0

which is solvable for arbitrary g ( t ) E C[O, 11. Yet, as in the discussion of (1.2.6) there can be no a priori estimate for a solutiony of (1.2.13) in terms of g since for g = 0, (1.2.13) possesses solutions u ( t ) of arbitrary amplitude, i.e., for which supLo,,] lu(t)l can be arbitrarily large.

(v) Resonance phenomena An interesting nonlinear effect related to small divisor problems is associated with the term resonance phenomena. These phenomena are well illustrated by the problem of “normal modes” for a nonlinear Hamiltonian system near a point of equilibrium. A system of N coupled linear oscillators is governed by the system of linear ordinary differential equations x +Ax =0 where ~ ( t€RN. ) (1.2.14) Here the matrix A is assumed self-adjoint and nonsingular with eigenvalues 0 < A: < A; < . < A;. Such a system has N linearly independent periodic solutions x j ( t ) with minimal periods 2r/Aj ( j = 1, . . . , N ) , called “normal modes.” These solutions can be found explicitly by diagonalizing A , so that (1.2.14) becomes uncoupled. Moreover, every solution of the system (1.2.14) is a superposition of these fundamental normal modes. It is of great importance to study the behavior of these normal modes (near

24

I

BACKGROUND MATERIAL

x = 0) under a nonlinear Hamiltonian perturbation VV(x), where I VV(x)l = o(lx1) for 1x1 sufficiently small. Thus, the new nonlinear Hamiltonian system can now be written (1.2.15) x + A x + VV(x) = 0. Following the Cauchy majorant method it is customary to seek solutions x(t) with period A near thejth normal mode xj(t) in the form x(s) =

ex,($)

M

+

an(s)en, n=2

where ?is = t and

e

is sufficiently small, m

A

= 2Ir/Aj

+2 n= 1

However, in order to prove that the resulting power series converges, it is necessary to make a severe assumption on the eigenvalues Aj, namely A k / A j # integer (for k = 1 , 2, . . . , N with k # j). Indeed, for any integer n , terms of the form (A, - nAj)-' appear in the formal series expressions for the coefficients a, and p,,. These irrationality conditions or resonance conditions thus seem essential for the permanence of thejth normal mode under nonlinear perturbation. This is indeed the case for non-Hamiltonian arbitrary perturbations. However, by restricting our class of perturbations to Hamiltonian ones (as defined above) a quite different fact is true; viz. these irrationality conditions are unnecessary for the preservation of normal modes. This interesting fact will be described in Chapter 4 as a bifurcation phenomenon. See Fig. 1.4 for an illustration of this situation.

FIG. 1.4 Illustration of the preservation of periodic motion of a normal mode under a Hamiltonian perturbation versus nonpreservation under a general perturbation.

1.3 FACTS FROM FUNCTIONAL ANALYSIS

25

1.3 Facts from Functional Analysis

Many of the problems concerning nonlinear systems mentioned so far can be reduced to the solution of an infinite number of equations (albeit nonlinear) in as many unknowns. Thus it is natural to attempt to extend the basic concepts of the functional analysis of linear operators to this broader context. Here we summarize those basic concepts and results from classical functional analysis needed in the sequel. Essentially, the elementary facts we need in the sequel concern (i) geometrical properties of Banach and Hilbert spaces, (ii) properties of bounded linear functionals and operators on a Banach space, (iii) facts concerning compactness in Banach spaces, and (iv) explicit examples of (i)-(iii) for certain standard Banach spaces. For complete proofs and references of the results mentioned here, see the bibliographic notes at the end of this chapter. 1.3A

Banach and Hilbert spaces

A Banach space ( X , 11 11) is a normed vector space (over the real or complex numbers) that is complete with respect to the metric d ( x , y ) = IIx - y 11. In the sequel, we shall be concerned primarily with such spaces and the (geometrically simpler) special case of Hilbert spaces. Recall that a Hilbert space H is a vector space with a positive-definite inner product ( , ) that defines a Banach space upon setting l\xl12= (x, x) for x E H . Therefore, in a Hilbert space H , orthogonal vectors and the orthogonal complement M I of a subset M of H ( M = { x I x E H , ( x , y ) = 0 for all ,y E M } ) are defined as in the finite-dimensional case. (Strong) convergence (or convergence in norm) of a sequence { x,} to x in a Banach space X means that (Ix, - XI] -0 as n+ 00. Thus, for example, f ( x ) = llxll is a continuous function with respect to convergence in norm. A seminorm defined on a Banach space X is a nonnegative real-valued functional 1x1 defined on H that satisfies the properties lax( = (a1 1x1 and ( x + yI < 1x1 + Jy(A . seminorm f is compact relative to 11 11 if every bounded sequence in X has a convergent subsequence in 1 I. Closed linear subspaces of Banach (Hilbert) spaces are again Banach (Hilbert) spaces. A similar result holds for finite direct sums; and if X is the direct sum of XI and X,,we write X = X , CB X , . However the rather intricate geometry of general Banach spaces can be seen by noting that there are closed subspaces M of a Banach space X for which no closed subspace N of X exists satisfying X = M 63 N . Fortunately, that situation does not occur if (i) X is a Hilbert space, (ii) dim M < co, or (iii) codim M < 00. In fact, if XI is any closed subspace of a Hilbert space X , then X = X , 63 X t .

26

1

BACKGROUND MATERIAL

If a normed vector space X has two norms 11 [I1 and 11 112, we call these norms equivalent if there are positive constants a and p such that aIIxIIl < J1xI12< P I I x I I , for all x E X. Much simplification generally results from a judicious choice of equivalent norms in the problems we shall discuss. The pleasant geometric properties of Hilbert space may be attributed to the so-called parallelogram law. In fact, a Banach space (X, )I 1)) is a Hilbert space if and only if the parallelogram law holds, i.e., for each u, u E

x,

I I u + v1(’ + I I u - 1111’ = 2( llull’+ I l ~ l l ’ } . This law can be generalized to a useful Class of Banach spaces as in the next definition. A Banach space (X, (1 11) is called uniform& convex if for all E > 0 and u, v of norm 1 in X with I1u - u11 > E, there is a S = S(r) independent of u, u such that 1) 4 ( u - u)ll < 1 - 6. Such spaces show many of the useful geometric properties of Hilbert spaces. Thus, for example, (1.3.1)

(1.3.2) Let M be a closed convex subset of a uniformly convex Banach space X, and let u be a point of X - M. Then the distance d(u, M) is attained by one and only one point m E M.

We say that a Banach space Y is imbedded in X and write X =I Y if (i) the elements of Y are also elements of X and (ii) (strong) convergence of a sequence { u,) in Y also implies (strong) convergence of { u,,) in X. This implies the existence of an absolute constant c > 0 such that [lullx < c(IuIJ for each u E Y.. Y is called compactly imbedded in X if in addition to (i) and (ii) bounded subsets in Y are compact in X. In many problems of analysis it is useful to consider those oneparameter families of Banach spaces X , where the parameter a varies over the positive integers or real numbers with the property that Xa,c X,, for a2 < a,. Such families are called scales of Banach spaces, or a Banach scale. A metric space is called separable if it possesses a countable dense subset. In most of the specific problems we consider, the Banach spaces involved are in fact separable. Linear subspaces of a separable Banach space X are separable, as are quotients of X by closed linear subspaces. Any separable Hilbert space possesses a countable orthonormal basis and consequently all such spaces are isometric. 1.36 Some useful Banach spaces

Let S2 be a domain in RN.Then the following special Banach spaces will prove important in the sequel. (I) Spaces of continuously differentiable functions Let m be a

27


nonnegative integer and a a multi-index. Then

cm(G) = { f I DYis continuous on il for ( a (< m } is a Banach space with respect to the norm

IIfIIm =

2

SUP

lal 1) for arbitraryf E Co(a). This difficulty can be overcome by defining (11) Spaces of Holder continuous functions Let a be a positive number, 0 < a < 1. Then a function u ( x ) is said to satisfy a Holder condition with exponent a in 52 if

Ha(u) = sup x,y€Q

-

lx

u(y)l

-A"

< 00

(0 < a

< 1).

The set C m . a ( a ) =( u 1 u E Cm(G),H a ( D B u )< co, 1/31 = m } is a Banach space with respect to the norm

Hence for a = 0, the norms I l f l l m , o and llfllrn are equivalent. This fact is useful if one notes that for fixed m,the space C'"*"(G)form a Banach scale as a varies over [0, 11. These spaces solve the potential theory problem mentioned above since iff E Car"(a),the Poisson equation Au = f always has a solution u E C 2 , a ( a )with 0 < a < 1. (Ill) Spaces of p-Integrable functions Let (52, 9, p ) be a measure space defined on 52, and let p be a positive number with 1 < p < 00. We denote by

[1 L

L,(% p ) = f

lflP

dp

I

< co,fp-measurable .

If one identifies functions differing only on a set of p-measure zero, then Lp(52,p ) is a Banach space with respect to the norm

28

I

BACKGROUND MATERIAL

Clearly L,(8, p ) is a Hilbert space with respect to the inner product (f,g ) L * = lnfg dP. With llfll L , = ess S U P n IfL

L,(Q, P I = {f I llfll, < ~ J P -meas u ra b le ) is a Banach space. Generally, we shall suppose (Q, 3 , p ) is the Lebesgue measure space defined on 8, and we write dp = dx. The relations among the Lp spaces, as the number p varies, will play an important role in the sequel. In particular, we note the following three important inequalities: (a) Holder’s inequality If f;, E Lp,(8,p) and 1 = is p-integrable and

Hence for p ( 8 ) < co andf

Ilfllr,

I

E L,(& p), -1-

(1.3.4)

x;”=1/ p i , then rIf=,f ;

G [P(8)lr

P

-I

IlfllL,

for r

< p.

Thus for p(Q) < 00, the Lp(8,p) spaces form a Banach scale for p a). (b) Moreover, if f E Lp n Lp+p for p > 0, then f E Lp+,p(Q,p) for 0 < t < 1 and $(s) = log llfl14 is convex in s for s E [ p , p + p ] . (c) Clarkson’s inequalities Let f, g E LP(& p) with p-I + q - l = 1, then for p >, 2:

E [I,

(1.3.5)

II f (f+ g>ll’., + II t (f- g>ll$ G 4 { Ilfll‘.,

+ II gll$).

An easy immediate consequence of (1.3.5) is that for 2 < p < co, L,,(8, p) is uniformly convex. An inequality, analogous to (1.3.5) also holds for L,,, 1 < p < 2, so that L,,(8, p) is also uniformly convex for 1 < p < 2. (Iv) (Soboiev) spaces of functions with generalized Lp derivatives In many problems involving differential operators, it is convenient to incorporate the Lp norms of the derivative of a function into a Banach norm. To accomplish this, consider the functions in the class Cm(Q). For any number p > 1 and integer m > 0, we take the closure of C m ( 8 )with respect to the norm

The resulting Banach space is called the Sobolev space Wm,p(Q).Now W,,,(Q) = Lp(Q), for fixed p , the Banach spaces Wm,,(8) are uniformly convex for 1 < p < 00 and clearly form a Banach scale as m varies over the nonnegative integers. Furthermore, for p = 2, Wm,,(a) is a Hilbert space


29

with respect to the inner product

x

JD“u.D”o

(u, u ) ~ , ~ =

lal<m

dx.

Q

The spaces Wm,,(a) can be modified to incorporate boundary conditions. Thus the closure of CT(3) in Wm,p(3),denoted Wm,p(3), contains functions whose derivatives up to order m vanish in the “generalized sense,” on the boundary a 3 of 3. If 3 is a bounded domain, a basic inequality of Poincare implies the existence of an absolute constant k ( 3 ) such that for ri.,,,(3) (1 < p < m), (1.3.7)

IIuIlLp

k(3)ll WLp.

Consequently, for bounded domains in Wm,p(3),the “short” norm given by

is equivalent to the norm given by (1.3.6). This is easily seen by iterating (1.3.7). The Sobolev spaces Wm.pcan be defined on Riemannian manifolds ( 9RN, g ) by using local coordinate neighborhoods and partitions of unity. g) can be defined as the closure of C For example, the space W , ,2( “X.”’, ( 9RN, g ) in the norm

where in terms of local coordinates we have

All these integrals are independent of the local coordinate neighborhoods and partitions of unity used to define them. The Sobolev spaces Wm,,(3)can be extended to cover “negative” orders of differentiation. If m 2 0, 1 < p < 00, and p - ’ + q-’ = 1, then taking derivatives in the distribution sense, W-m,p(a) =

{

u

Iu=

x

Dugat

ga

E~

IaI 6 m

and is a Banach space with respect to the norm IIuII-m,p

= sup

2 lal<m

I(

- l ) ’ ” ’ g aD ” f x

Q

over the class of functions f in Wm, with norm 1.

p ( 3 )

30

I

BACKGROUND MATERIAL

The Sobolev spaces are important in potential theory due to the property that if any f E L,(Q), 1 < p < 00, the Dirichlet problem for the equation Au = f has a “generalized” solution u E

e2,,(Q).

1.3C Bounded linear functionals and weak convergence

A bounded linear functional h ( x ) defined on a Banach space X is a linear mapping X + R ’ such that Ih(x)l < Kllxllx for some constant K independent of x E X. Clearly h ( x ) is continuous with respect to convergence in norm. Furthermore, the set of all bounded linear functionals on X, denoted X * is called the conjugate space of X. It is a Banach space with respect to the norm IJhlJ= sup lh(x)I over the sphere I(x(Ix= 1. Bounded linear functionals on X are also written x*(x), where x * E X * and x varies over X. If (X*)* = X,then the space X is called reflexive, and such spaces share many special geometric properties with Hilbert space. In particular, all uniformly convex spaces are reflexive. The following well-known results concerning the extension of bounded linear functionals will be important later on in the text.

(1.3.8) Hahn-Banach Theorem Let p ( x ) be a seminorm defined on a Banach space X, M a linear subspace of X, and f ( x ) a linear functional defined on M with If(x)l Q p ( x ) for x E M. Then f can be extended to a bounded linear function F ( x ) on X with IF(x)I < p ( x ) . This result has the following consequences. (1.3.9) (i) Let h ( x ) be a bounded linear functional on a closed linear subspace M of X. Then h ( x ) can be extended to a bounded linear functional F ( x ) on X with (lhllw = IIFllx. (ii) If xo E X, then there is a linear functional h E X of norm one with h(x,,) = ((xoll.Thus I I X I ( ~= sup h ( x ) over the unit sphere Ilhllx. = 1. (iii) Let u be a nonempty convex open subset of X, and M a linear subspace not intersecting u. Then there is a closed proper subspace of X containing M and intersecting u. Furthermore, it is useful to have a concrete representation for an arbitrary linear functional on a given Banach space X. In this connection, we note (1.3.10) (i) Rlesr Representation Theorem Let X be a Hilbert space. Then any bounded linear functional h ( x ) defined on X can be uniquely written as h ( x ) = ( x , y ) for somey E X. (ii) L,*(Q,p ) = L,(Q, p ) for 1 < p < co and p - ’ 4-l = 1. Thus, for such values of p, L,,(Q, p) is reflexive.

+

1.3

(iii)

FACTS FROM FUNCTIONAL ANALYSIS

6'z,p(Q)= W-,,,q(Q),m

an integer, 1 < p < 03, and p - l is reflexive.

31

+ q-'

= 1. Thus for such values of m and p , @,,,p(Q)

In order to discuss various types of convergence in a Banach space X, the bounded linear functionals on X prove exceptionally useful. A sequence x, E X converges weakly to an element x E X if h(x,) + h ( x ) for each h E X*. Weak convergence is novel only for infinite-dimensional Banach spaces X since it coincides with the norm topology when dim X < 00. Weak convergence has the following fundamental properties. (1.3.11) (i) Weak limits are unique if they exist (ii) If x, + x strongly in X , then x, + x weakly in X. (iii) If x , + x weakly in X, then { IIx,ll} is uniformly bounded and

IlxII

lim IIXnll.

(iv) If x,+x weakly, a convex combination of the x, converges strongly to x. (v) Let X be a reflexive Banach space, then X is (sequentially) weakly complete. (vi) In a uniformly convex Banach space X, x, + x weakly and llx,,ll+ llxll imply that x, + x strongly. In many nonlinear problems of analysis to be studied in the sequel, it is important to determine precise conditions for weakly convergent sequences to be strongly convergent. The simplest nontrivial result is (1.3.1 I(vi)) above. 1.3D Compactness

In discussing problems involving Banach spaces X of infinite dimension, one attempts to find subsets of X that share the main properties of closed bounded sets in finite-dimensional vector spaces. To this end, one says that a set M of a Banach space X is compact if M is closed (in the norm topology) and such that every sequence in M contains a strongly convergent subsequence. One then defines the notion of weak sequential compactness analogously. Results useful in this connection are: (1.3.12) (i) In a compact set M of a Banach space X, weak and strong convergence coincide. (ii) The closed convex hull of a compact subset of a Banach space is again compact. (iii) A bounded set in a reflexive Banach space X is weakly sequentially compact. (iv) If the sphere &,I = { x 1 llxll = n, x E X} in a Banach space X is compact, dim X < 00.

32

I

BACKGROUND MATERIAL

A mapping A with range a Banach space Y and domain a Banach space X is called compact if A(u) is relatively compact in Y for all bounded sets u in X. Such compact mappings have separable ranges since A ( X ) = A ( U ,“=,u,,) c U ,“=rA(u,). Since each set A(u,) is separable, A ( X ) is itself separable. A closer study of such operators is fundamental for our study and will be made in Sections 1.4E and 2.4. It is extremely important to determine specific criteria for compactness in the special Banach spaces introduced in Section 1.2. Two well-known criteria are:

(1.3.13) Arzela-Ascoli Theorem If 52 is bounded, then a set S of C(52) is conditionally compact if and only if the set S is bounded in the sup norm and the elements of S are equicontinuous. (1.3.14) M. Rlesz-Tamarkin Theorem If 52 is bounded and 1 < p < 00, then a set S of Lp(52)is conditionally compact if and only if (i) the set S is bounded in the Lp norm, and (ii) equicontinuous in the Lp norm + y ) - f ( x ) l l L p + O as Iyl-0 uniformly forf E S ) . (i.e.,. ( f I ] Immediate consequences of (1.3.13) and (1.3.14) that will be useful later are: (1.3.15) If 52 is a bounded domain, then bounded sets in the C”*a(52) norm are conditionally compact in the C””2a‘(52)norm, provided m 2 m’ and a > a’, with at least one of the inequalities proper. (1.3.16) Relllch’s Lemma If G? is a bounded domain, 1 I , then sets that are bounded in the T$‘m,p(52) norm are conditionally norm. compact in the Wm-,,p(52) The extensions of the above results to general unbounded domains

i2 c R N is important for many of the problems to be discussed. Some typical results in this direction are: (1.3.14’) The Riesz-Tamarkin theorem (1.3.14) is valid for unbounded domains 52 c R N provided we add to the conditions (i) and (ii): (iii) S is equismall at infinity (i.e., limR+ml l f l l 4 ( Q - ( x formly forf E S ) .

=0

uni-

(1.3.16’) The imbedding of W,,,p(52)+ W,,-l,p(52)is compact if and only if vol[i2 n { y 1 Iy - XI < I } ] + O as 1x1- 00. 1.3E Bounded linear operators

A linear operator L with domain X and range contained in Y ( X , Y Banach spaces) is bounded if there is a constant K independent of x E X


33

such that ))Lxll < Kllxllx for all x E X . Such operators are continuous with respect to both strong and weak topologies defined on X . The set of such maps for fixed X , Y is again a Banach space, denoted L ( X , Y ) , with respect to the norm IlLll = sup IILxll for I J X I / = ~ 1. Any bounded linear operator L E L ( X , Y ) has an adjoint L * E L ( Y * , X * ) that is uniquely defined by setting L*g = f,where f ( x ) = g(Lx) for every bounded linear functional f E X * . Thus J ) L * )=) 1 ) L ) ) ;and for two operators L,, L, E L(X, Y), ( a L , pL,)* = aL: + ,BLz and (L,L,)* = L:L:. The resolvent set p ( L ) of an operator L E L ( X , X ) is the set of all scalars X such that L - XI has a bounded inverse. All other scalars X comprise the spectrum of L , denoted a ( L ) . A number h E a ( L ) is an eigenualue of L if Ker(L - A I ) # 0 , and a nonzero element x E X is an eigenvector of L corresponding to the eigenvalue h if x E Ker(L - XI). The set of such eigenvalues is called the point spectrum of L . The essential spectrum of a bounded linear operator L, a,(L), consists of those numbers A E a(L) that cannot be removed from the spectrum by the addition to L of a compact linear operator C . It turns out that A s;I a,(L) is equivalent to the fact that XI - L has a closed range, finite-dimensional kernel, and cokernel with dim Ker(h1- L ) = dim coker(X1 - L). Some specific linear operators that play important roles in the sequel will be the following:

+

(1.3.17) Sobolev's Integral Operator Let f2 be a bounded domain in R N , and X a positive number. Then the linear operator defined by

has the following properties: (i) Forf E L,(Q) and X < N(l - l/p), S is a bounded linear operator ( ~ ) , p = min(1, N ( l - I / p ) -A). of L,(Q)+ C o ~ ~ where For X > N ( l - l/p), S is a bounded linear operator of L,(f2) (ii) .+L,(Q) for r < N p / ( N - ( N - X)p). (1.3.18)

Calderon-Zygmund

Singular Integral Operator

For

x E RN: let K ( x ) = w(x)/lxI", where ~ ( x is ) a positive C" function on R~ - (0) such that /,.x,=,w(x) dS = 0. The linear convolution operator Lu = K * u is then a bounded linear mapping from L,(RN)+ L,(R") for I

lal. I81< m

(where aa8 = aBa are bounded measurable functions) defined on a bounded domain 61 C R N taken together with appropriate boundary conditions on the boundary as2 of a. For the simplest case, the so-called Dirichlet boundary conditions D 4 1 , , = 0, In1 < m - 1, for U, u E C,oO(Q),we set

after numerous i?tegrations by parts. As before, by (1.3.39), there is a unique self-adjoint operator L E L( W,,,,,(a), Wms,(a)) such that {( u, u) = (Lu,u).

1.4 inequalities and Estimates

In order to carry out the program described at the beginning of Part I, one associates with a given nonlinear differential system S a mapping f(S ) acting between suitably chosen Banach spaces. The key point here is that in this association f(S ) should retain the major qualitative properties of the system S . These qualitative feature off( S ) include such properties as boundedness, continuity, and compactness, and can be established only by making essential use of inequalities and estimates of the type presented below.

40

1

BACKGROUND MATERIAL

We shall describe only two classes of results since they will be the main analytical facts necessary in the sequel. The first class (described in Sections 1.4A-1.4C) are the so-called calculus inequalities for the Sobolev spaces W,,,,,(D). These inequalities describe in precise terms the connection between the sizes of: (i) (ii) (iii) (iv) (v) Banach

the Lp norms of the generalized derivatives of a functionf, the Lp norms defined on a domain D c R N of the function f itself, the pointwise behavior of the function f and its derivatives, the dimension of the set D, and bounded sets in Wm,,(52) when regarded as subsets of “larger” spaces X 3 Wm,p(D).

The second class (described in Section 1.4D-1.4E) can be described as estimates for the solutions of linear elliptic partial differential equations of two types: (i) Lp estimates for ( 1 < p < co) (i.e., estimates of the solutions in an integral or “averaged” norm), and (ii) pointwise estimates in the Holder spaces C ” * “ ( n ) (0 < a < M ) (i.e., estimates in the usual pointwise sense). Remark on pointwise versus Integral estimates: The close relationship between the pointwise and integral estimates in both these cases is crucial for the resolution of the concrete nonlinear problems we shall discuss. Thus, for example, many nonlinear problems first written down in the classical pointwise form can be naturally reformulated and solved in a Hilbert space context. Then it is necessary to ensure that this “Hilbert space” solution yields a solution to the actual nonlinear problem as stated in terms of smooth functions. It is exactly at this point that the estimates to be described prove invaluable since the Hilbert space norms we utilize are inevitably integral norms and the estimates thus provide pointwise information to be obtained from integral data.

1.4A The spaces W,,,(O) (1

 N and N + pp < p imply u E C 0 , ” ( R Nand ) (1.4.2) I I U I I p . r ( p j < const.II V U I I ~ R N , ,

where the constant depends on the support of u but not on u itself; (ii) p < N and ( N - p ) r < N p imply u E Lr(RN)and (1-4.3)

l l ~ t l p I , IltullL, > a, while I(tEilll,p= 1, a contradiction. Thus the following result is of great importance: (1.4.7) Kondrachov Compactness Theorem and Extensions Suppose S is a set of functions in W , , p ( d " )with a common compact support

1.4 INEQUALITIES A N D ESTIMATES

43

and such that the set ( I l ~ l l , , ~E l uS ) is uniformly bounded. Then S is ) N + ,up < p , and (ii) Lr(RN)for conditionally compact in (i) C o , p ( R Nfor r < N p / ( N - p). In addition, in the exceptional case (iii) N = p , the functional ,fR. e k u is continuous with respect to weak convergence in S , while (iv) for r = N p / ( N - p ) the set S is not necessarily conditionally compact. Proof of (I): (i) follows immediately from the Arzela-Ascoli theorem, the boundedness of the Sobolev integral operator (1.3.33), and the fact that for bounded domains Cos"(0) c Cos"'(Q) is a compact imbedding for a' < a.

Proof Of (It): For W,,,,-L,, the result can easily be derived from the M. Riesz-Tamarkin theorem. For r in the open interval ( p , N p / ( N - p)), we use (1.3.35) and the logarithmic convexity property of the L,,norm stated in Section 1.3B. Indeed, if (f.) is any weakly convergent sequence in L?,(f,,}is strongly convergent in L,,, and settingp. = N p / ( N - p) for someO 0 and c2 are constants independent of u. Pro01 for m =: 2: The second order case m = 2 is easy, while the details of the more difficult > 2 case are given in many texts and will thus not be reproduced here.

m

2 J a n p ( x ) D% Dflu - R ( x , u, D"u) 14. IPI- 1 where R ( x , u, D%) is a bilinear form in u and D"u with (a1= I , with measurable, bounded coefficients. By virtue of Holder's inequality, we may suppose for any t: > 0 and some absolute constant c(c), that ( P ( x , D)u, U)Lz=

R ( x , 0%) €11 v 4120,2 - c(c>ll~Il;, 2' Consequently, the result follows from the uniform ellipticity of P ( x , D).

(11) Lp estlmates for solutlons of linear elliptlc equations Let L be an elliptic differential operator of order 2m defined on a bounded domain 3 c RN. For simplicity, we suppose the coefficients of L are C functions and the boundary of 3,a 3, is also of class C m. By a solution u of the elliptic boundary value problem (1.4.23) L u =f, Daulaa= 0

we mean a function u E Lp(C2)such that

(1.4.24)

s u L * q = Jfq

for all cp E Cc(3),

where L* is the formal adjoint of L (see Section 1.5).

(1.4.25) Theorem Let 1 < p < 00, and suppose u is a solution of (1.4. 23) in the sense (1.4.24), then u E W2m, and (1.4.26)

C cIIILuII0,p + C ~ I I ~ I I O , ~ ~ where the positive constants c,, c2 are independent of u. Furthermore, if Ker L = 0, then c2 = 0. IIuII2m.p

(111) C m * aestimates of Schauder type

(1.4.27) Theorem Under the same hypothesis for L and 3, suppose

1.5 CLASSICAL AND GENERALIZED SOLUTIONS OF DIFFERENTIAL SYSTEMS

47

u E C” “(Q) is a solution of (1.4.23) with f E C“ in the sense (1.4.24). Then u E C 2 m * a ( Qand )

(1.4.28)

I J u ~ I C ~ ~Q. ~c1{ Q , IILullc0.E + c2II~(IcoS*)-

idea of the proofs for L = A on RN: We consider Green’s function G(x,y ) for the Laplace operator A on RN: G b Y )=

(

- o ~ ’ ( N- 2 ) - ’ ( x - y ( 2 - N for -(2n)-’ In

JX

N > 2, for N = 2.

- y ~

We observe first that for f of compact support and Au = f, u = /w,G(x,

v)f(v) dY = h * J

so that (formally) ( 1.4.29)

D,DJu = (D,D,h)

*/.

Next we remark that the Calderon-Zygmund inequality (1.3.18) is applicable to this last convolution equation since for all N 2, DJh is smooth on RN - (0) and homogeneous of degree 1 - N, and hence is a Calderon-Zygmund kernel function. Thus by (1.3.18), we find that for any fixed p E (1, oo), IJu1Iz.p

const- Jlfllr, G const, JIAulIr,.

Next, to derive the Schauder-type estimate, we apply (1.3.19) to the equation (1.4.29) and again observe that the kernel D, DJh is a Calderon-Zygmund kernel function. Thus IluIJc~.* const. Ilfllo,n G const- IIAuIlo,~.

For the proof of the general case, the reader is referred to the papers by Agmon el al. (1959, 1964).

1.5 Classical and Generalized Solutions of Differential Systems

The notion of generalized differentiation allows one to distinguish between classical solutions of a differential system S and the solutions satisfying S in some averaged (i.e., integral) sense. Thus, for example, a locally integrable function u is a “distribution” solution of the linear ) = f on a domain differential equation of order m, Lu = ~ l a l = m u a ( xDau Q c RN if for all test functions cp E C?(Q), (1.5.1)

buL*cp=ifv,

where L* =

2

(-1)“ Daaa.

laJ=m

On the other hand, a “classical” solution of Lu = f on Q is generally a function u, rn-times continuously differentiable on Q, that satisfies Lu = f at each point x E 9. This broadened concept of a solution has proved extremely useful in the

48

I

BACKGROUND MATERIAL

theory of linear partial differential equations. Moreover, generally for linear elliptic operators L, the class of distribution solutions of Lu = f coincides with the class of classical solutions provided f and the coefficients a,(x) are sufficiently smooth. 1.5A Weak solutions in Wm,p

In general, distributions cannot be multiplied. Thus for nonlinear problems, another type of generalized solution, "weak solution" (intermediate between the notion of classical and distribution solution), is important. We say that u is a weak solution in Wm+ over the open set Q c R N of the Dirichlet problem (1.5.2)

2

@(u)=

(-l)'a'D"A,(x,u,.

. . , Dmu)=O,

DuuIan=O,

la16 m

if u E W,,,p(Q)and (1 S.3)

2 ( a (6

J A , ( ~ ,u, . . . , ~ p u ~) a nl

c = p

o

for all cp E c;(Q).

$2

Partial differential equations of the form (1 S . 2 ) arise naturally from problems in the calculus of variations as discussed in Section 1.1C. Integration by parts in (1.5.3) shows immediately that if a weak solution of & ( u ) = 0 is sufficiently smooth, it is a solution of a ( u ) = 0 in the classical pointwise sense (provided of course that the functions A , are smooth). Thus a classical solution of the Dirichlet problem for C? ( u ) E 0 is a solution in the weak sense. As is well known, the converse statement is false (even for linear equations). Example Let 0 = { x l x = (x,, x 2 , , , , , x N ) , 1x1 2). In D the Dirichlet problem for the equation (*) Au = ( N + 2 ) x , ~ , I x I - ~has the unique weak solution u ( x ) = x l x 2 log 1x1 E Wl, 2(Q). Since this weak solution u ( x ) is not continuous at x = 0, the Poisson equation (*) has no classical solution w ( x ) . For if w ( x ) existed, it would necessarily coincide with u ( x ) , since the only generalized harmonic function vanishing on as2 is identically zero (Weyl's lemma).

The notion of weak solution just introduced has proven highly successful in the systematic study of nonlinear differential equations since it conveniently splits the study of the solutions of such systems into two parts: one part concerned with the existence and nature of weak solutions, another concerned exclusively with the smoothness of such weak solutions. Furthermore, the structure of these weak solutions can generally be reformulated in terms of abstract operators (generally nonlinear) acting between suitable Banach spaces. Thus powerful results of functional analysis become applicable in the study of nonlinear differential equations.

1.5

CLASSICAL A N D GENERALIZED SOLUTIONS OF DIFFERENTIAL SYSTEMS

49

1.5B Regularity of weak solutions for semilinear elliptic systems

By enlarging the class of possible solutions for a system 5 ,extraneous and "unreal" solutions may be introduced into consideration. This contingency must be excluded. Thus any discussion of weak solutions must take up the problem of proving that such generalized solutions are sufficiently smooth to be solutions in the classical pointwise sense. Such questions are referred to as regularity theory. The simplest (and also one of the most useful) regularity theories involves boundary value problems for semilinear elliptic differential equations. Two key facts are used. First, the known generalized solution and the linearity of the top order part of the given differential operator are used to regard the regularity problem as one for a linear inhomogeneous equation with the inhomogeneous term in some Lp class. Secondly, the nonlinearity of the problem is used to iterate the gain in smoothness obtained from the linear regularity theory. By "recycling" the smoother generalized solution, u ( x ) say, back into the inhomogeneous term, we find (by virtue of the Sobolev inequalities) that this term is an element of some new L, class with r > p . This, in turn, yields an additional improved smoothness for u ( x ) via the linear regularity theory, and so on. The ideas used in studying this regularity theory can be clearly seen in the study of the following semilinear Dirichlet problem. Let L be a linear elliptic differential operator of order 2 m with smooth (say C ") coefficients defined on a bounded domain s1 c RN,so that

Then we suppose u E Wm,2(3) is a weak solution of the system (1.5.4)

Lu = f ( x , u )

(1.5.5)

Dnu)aQ = 0,

in 3 ; ) a )< m - 1.

By (1.5.3), this means that for all (1.5.6)

2

'p E

Cc(3),

/ a a a ( " ) D P u D"'p =

b/.I8lcm a

JI,f(x,

")QJ.

'

Thus, if f ( x , u ) is a C function of its arguments (say) and if one knows a priori that IuI < const. on 3, one can immediately conclude that u E c 2m. a . Indeed, u can be regarded as a weak solution of the linear inhomogeneous equation Lu = g(x) in 3 , where g(x) = f ( x , u ( x ) ) Thus by (1.4.25), u E W2m,pfor any 1 < p < 00. For sufficiently

E L,(O).

50

1 BACKGROUND MATERIAL

large p , by the Sobolev imbedding theorem, g(x) = f ( x , u ( x ) ) E C0*O1(52). Consequently, by (1.4.27), u E C2"' *(52). Hence one achieves a regularity result in this case by an iterative argument based on the Lp and Schauder estimates for linear elliptic equations, combined with the Sobolev imbedding theorems (1.4.12). This argument can be sharpened as follows: (1.5.7) Suppose f ( x , u ) is a Lipschitz continuous function of x and u satisfying the growth condition

for IuI sufficiently large. Then any weak solution of (1.5.4) is a classical solution in 52 and at all sufficiently smooth portions of 352. Conversely, if u > ( N 2 m ) / ( N - 2m), (1.5.4) may possess weak solutions that are not continuous in 52.

+

We simplify the proof of this result by supposingf(u) = k(u') and postpone justification until Section 2.2, where a study of the simple composition operator f(u) = f ( x , u ) is given. To prove the second part of the theorem, we let L = A, the Laplace operator, and observe that if r = 1x1 and 52 = ( x I 1x1 < I), then re E W , , , ( Q ) for a > 1 - N/2. Moreover, a simple computation shows that u = r' satisfies the equation (except at x = 0)

Au

+ K ( a , N)u("-*)/"=

0,

where K ( a , N) = 4 2 - a - N).

Thus for 0 > a > 1 - N/2, u = r" - 1 will be a weak solution in 6'1,$2) of the Dirichlet problem for an equation of the type (1.5.4), and the resulting nonlinear term f ( x , u) will satisfy a growth condition (1.5.8) with a = (a - 2)/a = 1 - 2/a > ( N + 2)/(N - 2). Finally, we observe that at x = 0, u = r" - 1 has a singularit9 and so is not continuous in 52. We return to the first part .and assume f(u) = kua with a < ( N + 2m)/(N - 2m). Then to show the weak solution u E Wl, ,(a) is actually smooth enough to be a classical solution, we use an iterative argument called a bootstrapping procedure; i.e., we gradually increase the regularity properties of u ( x ) by first showing that for any compact subdomain 0' of 51, u E,,,,(0') for some p and then show that u E WZm,,, for any finite j . Then as soon as N < 2mj, Sobolev's imbedding theorem implies u E Co, Thus f ( x , u(x)) E Co,,(52'). Consequently, the Schauder regularity theorem (1.4.27) implies that u E Czm,,(0'). In case N < 2m, the argument is easy since the Sobolev imbedding theorem implies that (u1' E L,(n') for any finite p. Therefore, since u ( x ) is known, we can regard the equation Lu = ku"(x) as an inhomogeneous linear elliptic partial differential equation in u ; whence the Lp regularity theorem (1.4.25) implies u ( x ) E W,,,,(Q') for any finitep'. In case PI > 2m, we first show that u E Wzm,p(52')withp = 2N(1 + r)/(N 2m) fqr some r > 0. To this end, we first observe that by the Sobolev imbedding theorem u E Wm,@) implies u E Lp(52)for p = 2N/(N - 2m). Hence kluJO E L,(52) for s = p / a . Since a < (N 2m)/(N - 2m), for some c > 0, s = 2N(l + c)/(N 2m). Then, as in the preceding paragraph, we may consider the equation Lu = k ( u " ) as a linear inhomogeneous elliptic equation for u and again (1.4.25) implies u E W2,,,, ,(0')with s = 2N(1 + r ) / ( N 2m). Now we show that u E WZm,J52')with s1 > s. Since u E Wzm,8(0'),the Sobolev imbedding theorem shows that u E L,,(O') for pI = Ns/(N - 2m). Thus kJul' E L,, for sI= p l / a Now to show that u has improved regularity, we note that

+

+

+

+

1.6 MAPPINGS BETWEEN FINITE-DIMENSIONAL SPACES

51

After a short calculation, we find that s,/s = (1 + c)(N - 2 m ) / ( N - 2 m - 4mr) > 1 + P. Consequently, the Lp regularity theorem for inhomogeneous elliptic equations implies not only that u E s, with sI > s, but also that after a finite number of repetitions of this last argument, u E W2,,,, for ? arbitrarily large. Thus the desired result is attained. The equation (1.5.4) is semilinear since the operator L is linear. The regularity theory of quasilinear elliptic equations of the form (1.5.2) is much more difficult unless either m = 1 (Le., for second-order equations) or N = 1 (i.e., for ordinary differential equations). For second order ordinary differential equations, the following regularity result suffices for many applications. (1.5.9) Suppose p cp E

e,Ja,

> I and

u(x) E

el,p ( a , 6) satisfies the following integral identity for all

b)

where F ( x , y , z ) is a C 2 function of its arguments. Then if F,,(x,y, z ) # 0,u ( x ) E C2(a,6).

Proofi By a simple application of Sobolev’s inequality, we note that the function j,XF,(x, ii, GX) = G ( x ) is Lipschitz continuous. Integrating the second term in the braces in (1.5.10) by parts we find

Now, 5, is an arbitrary bounded measureable function with J:{, possible redefinition of C ( x ) on a set of measure zero) (1.5.11)

dx = 0, so that (after a

Fz(x,ii, fix) = G ( x ) + const.

Since F,, > 0, the finite-dimensional implicit function theorem can be used to solve G,(X) in terms of G ( x ) and G(x), so that ii, is Lipschitz continuous. Hence G ( x ) must be continuously differentiable, and again (5.1.1 1) implies that iiz, is continuous, as required.

The regularity results for quasilinear second-order elliptic partial differential equations are quite difficult to establish and need not detain us at this stage. In the first place these results are the principal object of study of a number of excellent recent monographs (see the bibliographic Notes at the end of this chapter) and secondly they are not needed for the major part of our study. Indeed, by and large, most of the nonlinear problems in mathematical physics and differential geometry that we shall discuss involve only semilinear equations, for which simple results analogous to (1.5.7) suffice.

1.6 Mappings between Finite-DimensionalSpaces

A large portion of the study of linear systems is based on knowledge derived from the theory of finite-dimensional vector spaces and linear mappings between them. Consequently, it is natural to base a study of general (nonlinear) systems on those ideas that extend from linear algebra to a nonlinear context. In this section we mention some results in this

52

I

BACKGROUND MATERIAL

connection needed in the sequel. References for a full discussion and proofs will be given in the bibliographic notes at the end of this chapter.

1.6A

Mappings between Euclidean spaces

Let Q denote an open set in R N ,and f a smooth mapping (of class CP say): 52 +R‘. Then one attempts to determine the mapping properties off by studying the derivative off, f’(x) (i.e., the N X M matrix ( D j J ( x ) where f = (fl, . . . ,fn)).Thus, if at a point xo, rank(f’(xo)) = M , f maps a small neighborhood of xo onto a small neighborhood of f(xo). Such a point xo is called regular with respect to f . The complement in Q, i.e., the set t? = { x I x E Q, rankf’(x) < M } , is called the critical set and a point x E $2 is called a critical point. The set C? is closed in Q since if x, + x in Q, rank f’(x,) > rank f ( x ) . The following additional results concerning the set C are important. (1.6.1) Let 52 be an open subset of

RN.Then:

(I) Sard’s Theorem If f(x) is a p-times continuously differentiable mapping of Q into R m , then the critical values f(C?) have measure zero in R m , provided N - m + 1 < p. (11) A. Morse’s Theorem If F ( x ) is a N-times continuously differentiable real-valued function defined on Q, and C? denotes the critical points of F(x), F(C?) has measure zero in 52’.

These two results have numerous applications in analysis, and in Chapter 3 we shall discuss infinite-dimensional analogues of these facts. For complex analytic mappings f defined on a bounded domain Q of C into C ‘, many additional mapping properties off are known. Thus (1.6.2) (i) For N = M , then zo is a singular point off if and only i f f is not one-to-one near zo. (ii) If zo is a point on the set S = { z 1 f(z) = p ) in a small neighborhood U of ,z, S n U consists of a finite number of irreducible components { V i } each of which is either a point or contains an analytic V, contains an analytic curve not (nontrivial) curve. Moreover, if Vi # contained in 5. (iii) If S = { z 1 fI.

f(x)=p

a,

This sum is finite by virtue of the compactness of and the inverse function theorem. (ii) Suppose now that f is known to be a C ' mapping of Q+RIWN. Then by (1.6.1(i)) we can find a sequence of regular points ( p , } (with respect to ( f , Q)) such that pfl+ p in RN.We then define the degree off at P as d(f,p , Q) = d(f, Q).

;i%

pf13

(iii) Finally, if f is only known to be continuous in 52, there is a sequence of C mappings f,, +f uniformly on and we set

'

a,

The function d ( f , p , Q) can be shown to be well defined in (ii) and (iii), and the limits exist and are independent of the approximating sequences. The basic properties of the degree function follow readily from this definition. For a bounded domain D c RN,they are (1.6.3) (i) (boundary v a h e dependence) d(f, p , D,) is uniquely determined by the action of f ( x ) on aD. (ii) (homoropy invariance) Suppose H ( x , t ) = p has no solution x E aD for any t E [0, 11, then d ( H ( x , t),p D ) is constant independent of t E [0, 11 provided H ( x , t ) is a continuous function of x and r . (iii) (continuity) d( f , p . D ) is a continuous function of f E ~ ( 0 ) (with respect to uniform convergence) and of p E D. (iv) d(f, p , D ) = d(f, p', D )for p and p' in the same component of RN - f(aD). (v) (domain decomposition) If { D l } is a finite collection of disjoint open subsets of D andf(x) # p for x E (0u,D,), then d ( f , p , D ) =W

(

f

9

p3

Dl).

(vi) (Cartesian product formula) If p E D c R", p' E D' c R" and f : D + R " , g: D'-+R", then d ( ( f , g). ( p , p ' ) , D x D')= d ( f , p , D). d( g, p', D')provided the right-hand side is defined. (vii) If f(x) # p in 0,then d(f, p , D )= 0. (viii) (odd mappings) Let D be a symmetric domain about the origin, and f ( - x ) = - f ( x ) on d D , with f : D +Rn, and f ( x ) # 0 on aD, then d(f,0, D )is an odd integer. (ix) If d(f,p , D ) # 0, then the equation f ( x ) = p has solutions in D.

54

1

BACKGROUND MATERIAL

(x) Let f be a complex analytic mapping of a neighborhood U of the origin of @" into itself with f(0) = 0. If the origin is an isolated zero of f, and the Jacobian determinant det IJ,(O)l = 0, then d ( f , 0, U ' ) 2 2 for any sufficiently small open neighborhood U' of the origin. The degree of a mapping just described is most useful in establishing qualitative properties of continuous mappings. As a simple example, we prove (1.6.4) Brouwer Fixed Point Theorem Let f be a continuous mapping of the unit ball u = (XI 1x1 < 1) in R N into itself. Then f has at least one fixed point in u. Proot: We showf has a fixed point either on the boundary of a or in the interior. Supposef has no fixed point on the boundary aa of a, then the degree 6 = d ( x - f ( x ) , 0, 1x1 < 1) is defined. In this case we show 6 = 1, by use of the homotopy h ( x , f)= x - ff(x), t E [0, I], joining the identity mapping to x - f(x). Since f maps a into itself, If(x)l < I ; and so the equation h ( x , r ) = 0 has no solutions on the boundary of a, 30. Indeed, if the equation had solutions on 30, f = 1 and f would, of necessity, have a fixed point on aa. Thus by the homotopy invariance property of degree mentioned in (1.6.3), 6 = d ( x , 0,1x1 < 1) = 1 (by definition). Thusf has a fixed point in a by (1.6.3(ix)) and so the result is established.

In Chapter 5 , we shall investigate the extensions of the degree to classes of mappings between Banach spaces. Moreover, this extension is used there to solve many problems of analysis. 1.6B Homotopy invariants

Homotopy groups Let M ( X , Y) denote the set of continuous mappings between the topological spaces X and Y. Two mappings f, g E M ( X , Y ) are called homotopic if there is a one-parameter family of mappings f, E M ( X , Y) depending continuously on t E [0, 11 and joining f and g, i.e., such that fo = f while f l = g. One easily verifies that homotopy is an equivalence relation, and so partitions M ( X , Y ) into a set of homotopy classes, written [ X , Y]. Then one attempts to obtain information about these homotopy classes and, if possible, to determine the classes [ X , Y] in terms of the topological properties of X and Y. Accordingly, an algebraic structure is introduced into [ X , Y]. Thus if X = S I denotes the interval [0,1] with end points identified, one fixes a base point yo E Y and restricts attention to mappings f E [ S ' , Y] with f(0) = yo. For two such mappings f, g E [ S ' , Y ] , one defines [f]. [ g] = [f .gl, where


55

It is easily shown that this operation satisfies the group axioms on the set [ S ' , Y ] with fixed base point yo. The group is called the fundamental group of Y relative to yo, and is written n l ( Y ,yo). More generally, the higher homotopy groups, n,( Y , yo), are defined by considering the n-cube I" (the product of n copies of [0, l]), the boundary aI" of I", and the homotopy classes of mappings [ I " , Y ] that map aZ" into fixed base point yo. For two such mappings, one sets

f(21,,

t2r

. . , t") 1, t,, . . . , r,,) *

for 0 < t , < 4 ,

for f < t , < 1, g(2t, and again verifies that this operation makes the above homotopy classes into a group nn(Y,yo). The group nn(Y,yo)can also be described by considering the elements of [ S " , Y ] that map a fixed point so E S" intoy,. Indeed, if the boundary aI" of I" is identified with the point so, the quotient Z"/aZ" is topologically equivalent to S"; and so the elements of n,,(Y,yo) can be identified with the homotopy classes [S", Y ] that map a fixed point so E S" into yo. If Y is simply connected, nn(Y,yo) does not depend on the base point yo, so tha: the notation n,( Y ) is used. Of prime importance in homotopy theory is the calculation of nk(Sn), the homotopy groups of spheres. The following results are well known: (1.6.5)

nk(Sn)=(0)

for k

< n,

(1.6.6) r , ( S " ) wZ. The notion of degree of a mapping introduced in Section 1.6A can be used to refine (1.6.6) by giving an effective method _for determining the homotopy class of a given mapping f: S" -+S". Let f be any continuous exte_nsion off into (3 = { x 1 XI < 1 }. We then observe that the degree off is d,( f, 0, a). The following. result of H. Hopf shows that this degree is the only homotopy invariant of mappings between spheres of the same dimension. (1.6.7) Theorem Let f and g be two mappings of S" into S". Then f and g are homotopic if and only if the degrees off and g are equal.

) Next we consider the properties of the homotopy groups T , , + ~ ( S "for 00, the groups n n + p ( S " stabilize ) (i.e., T,,+~(S") x rn+I + p ( S n + lfor ) n sufficiently large). In fact, the following result of Freudenthal and Serre holds.

p 2 0. A fact of importance in Chapter 5 states that as n+

(1.6.8) Let p be a nonnegative integer. Then the homotopy groups r n + , ( S " ) are canonically isomorphic for any integer n > p 1. Moreover, for p > 0, these stable groups are finite.

+

In this connection, the notion of suspension of a mapping is crucial. To define this notion, let f be a mapping of S'-+ S", then the suspension off,

56

1

BACKGROUND MATERIAL

s(f>, is a mapping of S'+ + S"+ that is an extension off (provided we regard S' and S" as equators of S r + ' and S'+l, respectively). S(f) is constructed by continuously mapping the northern hemisphere of S r f into the northern hemisphere of S"+l, and similarly for the respective ' T S') southern hemispheres. Suspension induces a homomorphism E : ( + rr+I(S"+I ) and, in fact, this homomorphism yields the isomorphism of (1.6.8).

The unstable homotopy groups, T , , + ~ ( Sfor ~ )1 < n < p + 1 exhibit particularly interesting properties, and in fact their determination still poses rather deep topological problems. Hopf showed that r 3 ( S 2 )is infinite and is in fact isomorphic to the additive group of integers, while 7r4(S3)E Z, the Abelian group of two elements. In fact every group of the form ~ T ~ , , - ~ (for S ' )II an even integer is infinite, and consequently we conclude that much information is lost under the suspension operation for p > 0. Thus the utilization of these unstable groups in analysis is an interesting problem, which we shall discuss briefly in Chapter 5. The homotopy classification of restricted classes of mappings between infinite-dimensional Banach spaces is of great importance in the sequel. To illustrate this point, consider the problem of solvability for nonlinear operator equations. A natural method of approach consists in deforming a given equation to a simpler one in such a way that the solvability of the simpler equation implies the solvability of the given equation. This problem and its relation to infinite-dimensional homotopy will be discussed in Chapter 5.

1.6C Homology and cohomology invariants (I) Singular homology groups Let 8, denote the standard Euclidean simplex in W + ' . Then a singular p-simplex defined on a topological space X is a continuous mapping a of aP into X . A singularp-chain on E over an additive stbelian group C: is a 'formal linear combination c = x g l u i of singular p-simplices ui with coefficients g, in '3. The set of such chains C,(X, 9) forms an additive abelian group. I f f : X + X ' is a continuous mapping and c E C , ( X , ), then we can define an induced homomorphismf,: C,(X, G)+ C p ( X ' , L?) by settingf,(~g,u,)= ~ g , f ( u , ) . Now if Sp = [ x , , x2, . . . , xp] denotes the standard Euclidean p-simplex, then we define the boundary operator d on the associated singular simplex by setting d(x,,

XI,

*

. . , xp) =

P

(- I)"x,,

XI,

. . . ,i i , . . . ,xp],

i= 1

where a circumflex denotes the omission of that vertex. For a general


57

singularp-simplex a(x,, . . , , x), we set du = u,d(x,, x I ,. . . , x,); while for a general element a E C,(X, 6), a =Xgp,, we extend d by linearity so that da = E g , du,. Thus d is a homomorphism of C,(X, G)+ Cp- ,(A’,9 ) and d 2 = 0. The kernel of d : C p ( X , 9)+ C,-,(X, Y ) , denoted Z p ( X , G), is called the p-dimensional cycle group of X over d ; while the image of d : Cp+,(X,G), C,(X, G), denoted B p ( X , G), is called the p-dimensional boundary group of X over 5.Now the associated quotient group, denoted H p ( X , G), is called the p-dimensional homology group of X over G , i.e., (1.6.9)

H P ( X , G)= Z,(X, s ) / B , ( X , 9).

This definition can be extended by considering a subspace Y of X . Indeed, the homomorphism d: C p ( X , 4 )+ Cp- , ( X , d ) maps the subgroup Cp(Y , S)+ Cp-,( Y , 4 ) . Thus d induces a homomorphism dd: C p ( X , G ) / C p ( Y ,

9)- Cp- , ( X , G)/Cp-l( Y , G)

with dP-,dd = 0. Denoting the kernel of d, by Z p ( X , Y , G), and the image of d, by B,(X, Y , G ), we then define the pth relative homology group as (1.6.10)

H p ( X , Y , $ ) = Z,(X, Y , g ) / B p ( X , Y , 8 ) .

Clearly, (1.6.10) coincides with (1.6.9) if Y = 0 . The rank of the Abelian group H , ( X , A ) is called the 4th Betti number, R,(X, A ) , of the pair ( X , A ) : and the alternating sum x ( X , A ) = CF’,( - l)~R,(X,A ) is called the Euler-Poincare characteristic of ( X , A ) . In the sequel, it will be important to determine the homology of some well-known spaces. Thus if E N = { x 1 x ( < 1,x E R N ) and S N - ’ = a E N , we find:

Hp-1’

9)=

H,(SO, Y ) = G

0

if

4

if 4 = 0, N - 1 (for N # 1);

4f0,

N-1,

R.

(11) Slngular cohomology groups The singular cohomology groups of a topological space X , H P ( X , G ) relative to a fixed abelian group G can be defined from the associated singular homology groups, formally, by duality. Actually, the singular cohomology of X has an additional ring structure by defining a “cup” product between elements of the cohomology groups. In Chapter 6, we shall make use of this structure in

58

I

BACKGROUND MATERIAL

estimates of the number of critical points of a functional 9 ( u ) defined on an infinite-dimensional manifold X. (ill) Finite-dlmenslonal crltlcai polni theory of M. Morse The singular homology theory just defined is of fundamental importance in the Morse critical point theory of C 2 real-valued functions defined on a finite-dimensional smooth manifold. This theory begins by classifying the critical points of a C2 real-valued function F ( x ) defined on RN. The simplest critical points are called nondegenerate, and consist of the points xo at which V F ( x o )= 0 but, the Hessian determinant of F at xo, det IHF(xo)I, # 0. Such points are isolated and can, in fact, be classified by the dimension q of the vector space on which the quadratic form F"(xo)& & is negative definite for & E RN.This number q is called the index of the critical point xo, and a lemma of M. Morse shows that if x = 0 is a nondegenerate critical point of index q for a C 2real-valued function F(x), then there is a local coordinate system ( y , , . . . , y N )in a neighborhood U of 0 such that +

4

(1.6.11)

F ( y ) - F(0) = -

N

2 yz + i =2q + l y,?. i=l

More generally, if xo is any critical point of the C 2 real-valued function F ( x ) defined on RN, then xo is called degenerate if det IHF(xo)l = 0. If xo is a degenerate but isolated critical point of F(x), we may classify xo by its Morse fype numbers, i.e., by the number of nondegenerate critical points of various indices equivalent to xo. More precisely, the type number of an isolated critical point xo of a real-valued function F ( x ) E C2(RN),with F(xo) = co, is the sequence of positive integers (mo,m,,m,, . . . , ) defined by setting each

mq = Rq(FC0+' n O(xo), F'o-'

n

O(x,); Z)

(i.e., the qth Betti number of the pair (F'o'' n O(xo), F'o-' n O(xo)) relative to Z), where E > 0 is a sufficiently small number, F' = { x I F ( x ) < c}, and O(xo) is a sufficiently small neighborhood of x,,. The following facts are known about these type numbers. (1.6.12) The type numbers (mo, m,, m,, . . . , ) of an isolated critical point x o of a real-valued function F ( x ) E C 2 ( R N )are finite, and mq = 0 for

q > N.

(1.6.13) The type numbers of a nondegenerate critical point of index q are 0 for i # q, m,= 1 for i = q.

(

NOTES

59

(1.6.14) The type numbers of an isolated critical point xo of C2 realvalued function F ( x ) are lower semicontinuous functions of F in the sense that if G has only nondegenerate critical points on a small neighborhood U of xo, and G is sufficiently close to F in C ' ( U ) , then G has at least mq nondegenerate critical points of index q in U for q = 0, 1, 2 . . . . Furthermore, suppose Gn is a compact smooth N-dimensional manifold. Then the notions of nondegenerate critical point and degenerate critical point, index of nondegenerate critical point, and type number can be defined in terms of our definitions on an open set in RN by using local coordinate systems. Indeed, each of these notions is invariant under a local diffeomorphism. On such a compact smooth manifold %, it is easily proven (by Sard's theorem) that the family of real-valued functions F ( x ) that possess only nondegenerate critical points on % is open and dense in C2(%). Moreover, for any such function defined on %, the set 9RU = { x I F ( x ) < a ) is a deformation retract of Emb = { x I F ( x ) < b } , provided [b, a ] contains no critical level of F ( x ) . On the other hand, if F - ' [ a , b ] contains a single critical point of index q,LXb = 31t" u E q , i.e., 9Rb is homeoand a cell Eq of dimension q. morphic to the disjoint union of '-Ra Finally we mention an interesting and useful relation that holds between the Morse indices of a C2 real-valued function F ( x ) defined in the neighborhood of an isolated critical point x,, E R N and the Brouwer degree of the mapping! = grad F relative to a sufficiently small sphere uc centered at xo. Indeed, under rather general boundary conditions on au,, if M,(xo) = (mo,ml,. . . , mN), the following formula holds: N

(1.6.15)

d(f,xo,u,)=

2 (-l)'m,. 1=O

NOTES A Hlstorlcal note on systamatlc approaches to nonllnear problems ot ancllyrls

Nonlinear problems of analysis arose naturally with the advent of calculus. Explicit and ingenious methods of solution abound in the mathematical literature of the seventeenth and eighteenth centuries. This work led Euler and Lagrange to consider the general theory of the calculus of variations. Moreover, in attempting to put Newton's method of undetermined coefficients on a rigorous basis Cauchy was eventually led to the majorant method for analytic nonlinear problems. The widespread use of this method of proof persists to the present day. Cauchy also used minimization methods (the method of steepest descent) systematically in the study of the zeros of simultaneous algebraic or transcendental equations over the reals. However, a new dimension to our subject was added by Poiocare beginning with his thesis

60

1

BACKGROUND MATERIAL

in the 1870’s. Poincare focused attention on the qualitative aspects of nonlinear poblems and thus opened a whole new variety of questions for mathematical investigation. Motivated by a systematic study of physics and geometry, Poincare introduced new concepts in such diverse areas as bifurcation theory (a term that Poincare himself coined), the calculus of variations in the large, application of toplogical methods to the study of periodic solutions of systems of ordinary differential equations, to mention only a few. Hilbert’s well-known lecture at the International Congress of 1900 contained a number of intriguing nonlinear problems for analysis and in particular stimulated research on nonlinear elliptic partial differential equations. This last topic proved to be a decisive one for progress on a more abstract level. In particular, the results obtained by S. Bernstein in the early part of the twentieth century on Hilbert’s problems for nonlinear elliptic partial differential equations were sufficiently general to provide a basis for later abstraction and generalization. Somewhat earlier Picard introduced the idea of successive approximation into nonlinear analysis. This idea was a natural extension of the Cauchy majorant method and was subsequently extended by S. Banach in his thesis of 1920 to the contraction mapping principle. This paper marked the quiet birth of nonlinear functional analysis. Other key results of this period include E. Schmidt’s work on nonlinear integral equations and Liapunov’s study of bifurcation phenomena associated with rotating figures of equilibrium. A key paper in the development of nonlinear analysis was the paper “Invariant points in function space” by Birkhoff and Kellogg that appeared in 1922. This paper inspired much research on fixed point theorems in infinite-dimensional spaces, as well as other extensions of algebraic topology to analysis. The most penetrating research was due to J. Schauder, who applied his abstract results systematically to problems in nonlinear elliptic partial differential equations. This development reached a high point in 1934 when the paper “Topologie et equations fonctionelles” by Leray and Schauder appeared. Cf. Cacciopoli (193 I). As a final key development in the early study of nonlinear problems we mention the advances in the calculus of variations in the large made by Marston Morse beginning in 1922 and later by Liusternik and Schnirelmann. The Second World War virtually destroyed the Polish school of functional analysis, and the joint book on nonlinear problems of analysis planned by Banach and Schauder never appeared in print. B Sources of nonlinear problems In mathematlcal economics As an example of the type of nonlinear problems that arise in economics we mention the integrability problem that arises in the theory of consumer behavior. The empirical situation, assumed given, consists of a consumer with fixed income M acting in a commodity space of dimension n + I, where each commodity sells at a given price pi (assumed to be strictly positive), and demand functions

x j = . Q ( p I , p ~ ., . . , p n + l , M )

( j = 1, 2,

these functions uniquely determine the amount of thejth commodity selected x, as a function of prices and income. The integrability problem consists in determining conditions on the functions Q that ensure that the consumer acts in such a way that he maximizes a “utility function” subject to budget constraints. This problem is thus a simple example of an “inverse problem in the calculus of variations.” Moreover, after appropriate normalizations and reductions the problems can be studied via the solution of the nonlinear partial differential equation aM ( j = 1,. . . , n + 11, - = Q ( p , , *, . . . .P,,+~. M ) M ( p o ) = M,,. aPj A particularly interesting situation occurs when the demand functions are assumed Lipschitz

NOTES

61

continuous but not differentiable, a case that has useful applications in mathematical economics. For further discussion of the interesting history and recent developments in this area, the interested reader is referred to the recent book “Preference, Utility and Demand” edited by L. Hurwicz, J. Chipman, and others, and to the article by Berger and Meyers (1971) in that volume. C Dlmenslonal analysls and Integral Inequalities Many integral inequalities of Section 1.4 relating L, and Sobolev norms hold with an absolute constant independent of the associated domain P of R”. To find additional information about this case a simple device known as dimensional analysis is useful. The device consists in noting that if an inequality holds for a given function u ( x ) , it must also hold for u ( c x ) where c is a constant that can take any positive value. For example, suppose an inequality of the form

holds, where K is a absolute constant independent of the size of P a n d p is a positive number to be determined. The interested reader will easily show by dimensional analysis that the only value of p for which such an inequality can hold is p = 2 n / ( n - 2), a result totally consistent with (1.4.5).

D Weighted norms and the Kondrachov compactness theorems for unbounded domalns The Kondrachov compactness theorem (1.4.7) fails for general unbounded domains ( e g ,

RN), and as mentioned in the text, this loss of compactness is crucial for many interesting nonlinear problems. Thus, it is interesting to note that the Kondrachov compactness theorem can be extended to general unbounded domains provided appropriate weights (decaying at infinity) are introduced into the Sobolev norms. As a simple example, we mention the following result which will prove useful later in Chapter 6 : Let P be any domain in R“, then if ( uk) is a sequence of functions with uniformly bounded W,,$) norms, then ( u k / l x l “ ) has a convergent subsequence in Ll(Q) provided q > p > I , a is a number satisfying ( a - n ) / q < s - n / q , and m,p, and q are related as in Kondrachov’s theorem. For further results in this direction the reader is referred to the paper of Berger and Schechter (1972) in the References. E The Korteweg-Devrles equation

The interesting Korteweg-Devries equation u,

+ uu,

+ u,,,

=0

( x , 1) E R2

first arose in the approximate theory of water waves. It possesses traveling wave solutions of the form u ( x , t) = s(x - ct) for positive number c. Here s(x) = 3c sech2(x6/2) and is called a “solitary wave” or “soliton.” Moreover, it has been observed that any one solution of this equation vanishing as 1x1 + w can be regarded asymptotically as It1 + 00 as a superposition of a finite number of solitons. This equation also possesses an infinite number of integrals of motion, and in a certain sense is “integrable.” See the papers by Lax (1968) and Zakharov and Faddeev (1971) for further discussion. F Blbllographlc notes

Section 1.1: Early uses of qualitative nonlinear analysis in studying closed geodesics on manifolds can be found in Poincare (1905), Birkhoff (1927), and Morse (1934). Poincare’s conjecture concerning the existence of at least three closed simple geodesics on an ovaloid

62

I

BACKGROUND MATERIAL

has provided motivations for many deep researches on nonlinear problems beginning with Ljusternik and Schnirelmann (1930). The reader is referred to Courant (1950) for a historical discussion of Plateau’s problem. The generalization of this problem to higher dimensions has proved a remarkable achievement of recent years requiring the study of geometric measure theory, as in Federer (1969). This topic is however outside the scope of the present book. (See Nitsche, 1974.) The uniformization of algebraic curves via nonlinear partial differential equations, as described here, is discussed in Poincare (1890) (cf. Berger, 1969). A good discussion of the equation (1.1.6) concerning conformal metrics of prescribed Gaussian curvature can be found in Kazdan and Warner (1974). The paper of Yamabe (1960) has provided much of the impetus for modern research on the existence of metrics with prescribed curvature properties and the smooth solutions of nonlinear partial differential equations. The nonlinear aspect of the deformation problem for complex structures on higher dimensional complex manifolds is mentioned in Nirenberg ( 1964). A fine survey of nonlinearity in classical mathematical physics can be found in von Klrm6n (1940). It is remarkable that the abstract structures underlying these nonlinear problems have been so little studied. For further references concerning these topics we recommend the following recent books: Szebehely (1967) for celestial mechanics, Volmir (1967) for nonlinear plates and shells, and Batchelor (1967) for fluid mechanics. The latter book contains many fascinating pictures of vortex rings. Unfortunately the literature of contemporary mathematical physics is so diverse that unified discussions of the nonlinear phenomena appearing there are singularly hard to find. Our discussion of (1.1.21) is based on Wightman (1974). Einstein’s monograph (1955) clearly states the importance for relativity theory of new methods in finding singularity free solutions of nonlinear partial differential equations. Interesting discussions of phase transitions can be found in Titza (1960), Landau (1937), and Brout (1967). Section 1.2: Intrinsic properties of nonlinear systems are well discussed in Heissenberg (1967). See the articles of Landau (1944) and Ruelle-Takens (1973) for the relation between turbulence and the critical dependence of nonlinear systems on parameters. Interesting remarks concerning dimension and nonlinear growth can be found in von Neumann (1957). Historically speaking, studying nonlinear problems with methods developed for linear ones has proved the chief source of nonintrinsic difficulties. Thus the persistence of normal modes under nonlinear Hamiltonian perturbations as defined by equation (1.2.15) can be studied by the majorant method as in Liapunov (1892). However the results obtained are rather weak. A deeper investigation requires a combination of analytic and topological techniques, as in Berger (1970) and Weinstein (1974). Section 1.3: The material discussed in this section is relatively standard. General references for proofs include the following texts: Smirnov (1964), Riesz and Nagy (1952), Schechter (1971), Dunford and Schwartz (1958, 1963), and Yosida (1965). Section 1.4: Sobolev’s theorem was originally proved in Sobolev (1938). Its sharpening (1.4.1) is due to Nirenberg (1959) and Trudinger (1967). Proofs of much of the material of this section including the Calderon extension theorem can be found in Agmon (1965). Section 1.5: The reader will find an elementary and informative. discussion of weak solutions of elliptic boundary value problems in Sobolev (1950). The use of the bootstrapping procedure to obtain regularity for solutions of semilinear elliptic boundary value problems has been known for some time. The result (1.5.7) of the text was obtained in Berger (1965).

NOTES

63

Section 1.6: Proofs of the elementary results of differential topology discussed here can be found in Milnor (1963, 1965). For the results from algebraic topology, the reader is referred to the books Spanier (1966), Hilton (1953), and Wallace (1970). Finite-dimensional Morse theory and the Morse type numbers are well discussed in Morse (1934), Seifert and Threfall (1938), Milnor (1963), and Pitcher (1958).

CHAPTER 2

NONLINEAR OPERATORS

This chapter is divided into seven sections. In the first two sections we develop a calculus to deal with abstract nonlinear operators and show how concretely given operators can be reformulated in this abstract context. The next five sections take up the definition and properties of special classes of nonlinear operators. Each of these special classes will prove useful in the sequel. Two key ideas are used in the definition of these classes: first, defining a class of smooth nonlinear operators by use of the FrCchet derivative (i.e., by linearization); and secondly, defining a class by extending notions of mappings between finite-dimensional spaces.

2.1 Elementary Calculus

Many results of elementary calculus apply equally well to mappings between infinite-dimensional spaces. We now explore this important fact. We begin by fixing notation as follows: X and Y denote Banach spaces, and f denotes a given mapping from X to Y , and we write f E M ( X , Y ) . We shall discuss the following desirable properties for f : boundedness, continuity (with respect to various types of covergence), integrability, differentiability, and smoothness.

2.1A

Boundedness and continuity

The map f E M ( X , Y ) is called continuous (with respect to convergence in norm) if x, .+x in X always implies f ( x , > + f ( x ) in Y . f is said to be bounded if it maps bounded sets into bounded sets. f is called locally bounded if each point in the domain off has a bounded neighborhood N such thatf(N) is bounded. In th e case whenfis linear, the two concepts of continuity and boundedness are equivalent; but this is not true in general. Since continuous maps of a finite-dimensional Banach space X into a Banach space Y are necessarily bounded, one naturally seeks to extend this 64

2.1

ELEMENTARY CALCULUS

65

result to infinite-dimensional spaces. To accomplish this, we introduce the notion of uniform continuity of the mappingf. (2.1.1) A mappingf is uniformly continuous if for every c > 0 there exists a S(c) > 0 such that 1I.x -y(I < 6 implies Ilf(x) -f(y)ll < c. Clearly a uniformly continuous mapping is continuous. In fact, we have (2.1.2) A uniformly continuous mapping is bounded. Proof: It suffices to show thatf maps any sphere S, = (XI 11x11 < r } into a bounded set. For any c > 0, by the uniform continuity of f, there is a 6 > 0 such that I(x -yII < 6 implies Ilf(x) - f(y)l( < c for x,y E S,. Choose n to be any positive integer satisfylng n6 > 2r. Then if a,b E S,, there are n points x, E S, with J(x,- x,< 6 and x,, = a,x,- I = b. Hence

a number independent of the choice of a and b ; from which the result follows.

Actually (besides continuity with respect to convergence in norm) there are three distinct and important notions of (sequential) continuity for mappings f between general Banach spaces X and Y . These notions are obtained by considering the possible actions off on the weak as well as the strong topologies of X and Y . Thus a map f E M ( X , Y ) may (i) map strongly convergent sequences in X into weakly convergent sequences in Y , (ii) map weakly convergent sequences in X into weakly convergent sequences in Y , or (iii) map weakly convergent sequences in X into strongly convergent sequences in Y . This last continuity property (iii) is called complete continuity since it implies the other two. Property (ii) is called demicontinuity. The alternative notions of continuity are sometimes useful in proving boundedness of a map f independently of uniform continuity assumptions. In fact, we have (2.1.3) Let X be a reflexive Banach space and f E M ( X , Y ) . If f maps weakly convergent sequences in X into sequences weakly convergent in Y , then f is bounded. Proof: We argue by contradiction. Suppose there is a bounded sequence (x,) in X such that Ilf(x,,)ll+ co. By the reflexivity of X , (x,) has a weakly convergent subsequence {x,,,) (say). By hypothesis, ( f ( x n , ) ) is weakly convergent and hence, by (1.3.1 I): uniformly bounded. But this fact contradicts the fact Ilf(x,,)ll+ m.

In the sequel continuous mappings between Banach spaces X and Y are denoted C ( X , Y ) . 2.1 B Integration

Criteria for the integrability of Banach-space-valued functions can be defined by considering an associated one-dimensional integral. Suppose a

66

2 NONLINEAR OPERATORS

function x ( t ) is defined on a measure space (T, p, a( T)) with range in a Banach space X. Then a definition by duality is as follows. Definition x ( t ) is integrable if there is an element I,(x) E X for each element E of the a-ring a ( T ) such that (2.1.4)

= J x * ( x ( t ) ) dp

x*(Z,(x))

(in the Lebesgue sense)

E

for each x* E X*. We set / , x ( t ) dp = I,(x). Clearly the operator I, so defined is linear. The Hahn-Banach theorem then ensures that I E x ( t ) dp is well defined and that (2-1.5)

IIJ-(t) 441 <Jllx(l)ll E

4.

Thus x ( t ) is p-integrable if the operator I ( x * ) = I , x * ( x ( f ) ) dp is in X** and can be identified with the usual imbedding of X + X". Hence it is difficult to decide exactly what constitutes the class of integrable functions in the sense of this definition because even if x * ( x ( t ) ) is measurable and integrable for each x*, in general I,(x) E X * * . Note that, for reflexive spaces X the definition works well. In general, we have (2.1.6) Theorem Suppose x ( t ) is a measurable function on a measure space M = (T, p, a ( T ) ) with values in a Banach space X (in the sense that it is a step function in M or the limit (in norm) p-a.e. of a sequence of such step functions). Then x ( i ) is p-integrable if Ilx(t)ll is p-integrable. Prool: First we prove the result for countably valued functions. Suppose x ( f ) = xk on Ek for

k = 1, 2, . . . and JEllx(r)ll dp < 00. Set m

IE(x) =

2 xk k- I

p(Ek

n E),

where Ek is a p-measurable set. This series is absolutely convergent since

2

lIxklI

k-I

p(Ek

n

=JAlx(t)ll

dt

< 00.

Hence for any x * E X I , W

X*(IE(X))

=

X*(Xkj k- I

E)=

/-*(X(r)) 4.

Thus, in this case, I E ( x ) E X and I E ( x ) = j E x ( f )dp. Now suppose x ( r ) is an arbitrary measurable function with values in . 'A Then, by definition, there is a sequence of countably valued functions ( x N ( t ) )such that ( 8 ) IlxN(r)x(r)ll + O a.e. Suppose for the moment that in addition ( 9 8 ) JE-IxN(f) - x(f)ll dr+O. Clearly, Jlx(l)- x,(f)ll is measurable; and since I E ( x ) is a linear function for countably valued functions x ( t ) ,

67

2.1 ELEMENTARY CALCULUS

I E ( x N )= ( j E x N ( r )d p ) is a Cauchy sequence in X. This sequence posseses a limit X, which we define to be I , ( x ) . Clearly I E ( x ) E X and is independent of any approximating sequence ( x N ) satisfying(*) and (**) above. Now for' any x* E X*,( x * , x N ( f ) ) + ( x * , x ( r ) ) p - a.e. and also

Thus /Ax*. X ( I ) ) dp = lim / ( x * , x N ) dp = lim ( x * , I E ( x N ) )= (x*, I , ( x ) ) , N-m

N-m

E

Consequently x ( r ) is integrable. Finally, we construct an appropriate sequence x N ( r ) satisfying (*) and (**), provided jEllx(r)ll dp < m. Since x ( r ) is the limit of step functions, there is a countable sequence in X,11,12,. . . that is dense in the range of x ( t ) . Set w N ( f )= lk,where k is the smallest < l / N . The function w N ( f )is measurable and w N ( t ) + x ( t ) for all 1. integer with I x ( t ) Furthermore, since the composite of a continuous function and a measurable function is again measurable, 11 wN(r)II and Ilx(r)ll are measurable real-valued functions. Consequently, there is a sequence of simple functions such that 0 < g N ( x ( r ) ) G Ilx(r)ll and gN(r)+ Ilx(r)ll. Let x N ( f )= (gN(r)/llwN(r)ll)wN(r)for w N ( t ) # 0, and x N ( t )= 0 otherwise. Then x N ( t ) is countably valued, clearly measurable, and in addition IlxN(r)ll 4 Ilx(f)ll with xN(r)- x ( f ) . Since Ilx(t)ll is integrable, IlxN(r)ll is integrable. Also, I l x ( r ) - xN(f)II < 2llx(r)ll, so that by the Lebesgue dominated convergence theorem, jEIIxN(r)- x(r)ll dp-0. Hence ( x N ( f ) )is the desired sequence satisfying (*) and (a*).

(2.1.7) Corollary Let x ( t ) be a continuous function of [ a , b ] + X . Then is integrable and (as usual)

x(t)

pf)

N

df = W

with t,* E [t,,

+ m

z

x(t,*N&+, -

n=O

41 as It,+, - t n 1 + 0

["+,I.

Proot: The function x ( t ) is uniformly continuous on [a, b], and hence the step functions are dense in the Banach space of continuous functions C ( [ a ,b ] ; X ) .

For further results concerning integration we refer the reader to Gross (1964). 2.1 C

Differentiation

The two main notions of differentiability at a point for an operator

f E M ( X , Y ) are defined as follows: Definition f E M ( X , Y ) is Frechet differentiable at xo if there is a linear operator A E L ( X , Y ) such that in a neighborhood U of xo, Ilf(x> -f(xo) - A ( x -

x0)II

= o(llx - xoll).

In this case we write A = f ( x O ) ,andf'(xo) is called the Frechet derivative I The notations ( x * , y ) and x * ( y ) for bounded linear functionals on X are used interchangeably in this chapter.

68


off at xo. If the mapping x + f ’ ( x ) of X called C 1 at xo.

+=

L ( X , Y ) is continuous at xo,f is

Deflnltion f E M ( X , Y ) is Gateaux differentiable at xo if there is an operator df(xo,h ) E M ( X x X , Y ) such that

lim llf(xo + th) - f(xJ - f df(xo,h)ll = 0 t +o for (xo t h ) E U , a neighborhood of xo. Furthermore, df(xo, h ) is called the Gateaux derivative off at xo, and we write

+

Some obvious properties are: (2.1.8) The Frechet and Gateaux derivatives are unique. (2.1.9) df(xo, ph) = /3 df(xo, h ) for any scalar

p.

(2.1.10) The Gateaux derivative commutes with bounded linear functionals, i.e., if y* E Y* and f E M ( X , Y ) is Gateaux differentiable at xo,

(2.1.1 1)

I f f is Gateaux differentiable at xo

+ th (0 Q t < 1)

,

Indeed, by (2.1.10),

+

so that ( y * ,f ( x o th) - f ( x o ) ) = (y*,JAdf(xo + th, h)) dt by the definition of integration. Sincey* E Y* is arbitrary, the result is established. The following result can easily be proven by the interested reader. (2.1.12) The Frechet differentiability and derivative of a map f E M ( X , Y ) are defined independently of equivalent norms in X or Y . The relationship between Gateaux and Frechet differentiability is given by (2.1.13) Theorem If f E M ( X , Y ) is Frechet differentiable at xo, it is Gateaux differentiable at xo. Conversely, if the Gateaux derivative off at xo, df(x0, h), is linear in h, i.e., df(x,;) E L ( X , Y ) and is continuous in x as a map from X + L ( X , Y ) , thenfis Frechet differentiable at xo. In either case we have the formula f’(xo)y = df(xo,y ) .

69


Proof: The fact that Frechet differentiability implies Gateaux differentiability follows immediately from the definitions. To prove the converse, we first note that by the hypothesis and (2.1.9) above we can write df(x, h ) = d f ( x ) h , where d f ( x ) E L (X,Y ) and Ildf(x, h)ll < IIdf(x)ll llhll. Thus utilizing (2.1.1 1) above, llf(x

+ h ) -f(.)

-

df(x)hll = IlJ’idfCx 0

+ th, h ) - df(x, A ) } dill

(2.1.14) Maps with a uniformly bounded Frechet derivative are uniformly continuous, and hence continuous and bounded. Rules for Frechet differentiation are similar to the finite-dimensional case: (2.1.15) The Chain Rule Suppose X , Y , Z are Banach spaces and U c X , I/ c Y are open sets. Then i f f E M ( U, Y ) and g E M ( V , Z ) with

f - ’ ( v >c u,

t g f ( 4 l ’ = g ” f W .f’(X>. (2.1.16) Product Rule Let U c X be an open set and f E M ( U , R’), g E M( U, Y ) be differentiable. Then h ( x ) = f ( x ) . g ( x ) is differentiable and h’(x)y = f ’ ( x ) y . g ( x ) + f ( x ) .g‘(x)y. (2.1.17) Suppose U is an open set of X and f: U+ Y , where Y is the product space Y = IIy= I y.. Then i f f = ( f l , fi, . . . ,fN), where J;: U + Y, is differentiable, f is differentiable and f ’ ( x ) = ( f ; ( ~ )f,’(x), , . . . ,fN’(.x)). Proof of (2.1.15): Supposef(x + y ) = f ( x ) + f ’ ( x ) y + o(llyll), then

g.f(x +

v)= g(f(x) + /’(xlv = g(f(x)) = gf(x)

+

.(llull))

+ g’(f(x))[f’(xlr + o(llrll)l+ o(llull)

+ g’(f(x)).f’(x)y + ~ ( l l Y l l ) .

Proof of (2.1.16): Suppose f and g are expanded in the form f ( x

+y ) = f ( x ) +f’(x)y+

~ ( l l Y l l )Then ~ f(x + y M x + Y ) =f(x)g(x) + [f’(x)vlg(x> + [f(x>lg‘(x)v+ o(llull). The proof of (2.1.17) is obvious, being the same as in the finite-dimensional case.

Partial derivatives of a mapping f E C’(U, Y) can be easily defined :’, U, and each U, is an open subset of a Banach space X,. provided U = I Indeed, if x = (x,, . . . , . x N ) E U with xi E U,, the (Frechet) partial deriva-

70


tive off with respect to x,, D l f ( x ) , is defined by writing f ( x , , x2,

*

. . , X I + h. . .

*

9

XN)

-f(x1,.

*

f

I

XI,.

* *

9

XN) = P ( h )

+ o(llhll),

where P ( h ) E L(X,, Y), and setting D , f ( x ) h = P ( h ) , provided this expansion is valid. Clearly, D , f ( x ) E L(X,, Y) if it exists. Moreover, just as in standard calculus text books, we can prove that if D , f ( x ) E L ( X , , Y ) , N

(2.1.18)

2 D,f(x)h,.

f’(x)h =

r=l

The mean value theorem of elementary calculus has the following analogue for Frechet differentiation. (2.1.19) Theorem Suppose f E M ( [ a , b],X) is a Frechet differentiable mapping and l l ~ ( t ) l lG l’(t)for t E [a, b ] . Then (2.1.20)

Ilf(b) - f(.)l

(2.1.21)

IlfW -f(.)ll

P m t : Clearly for x * (2.1.11) above

G l(b) - {(a), SUP ( € [ a . bl

EXI,

Ilf’(l)llP - al.

the real-valued function ( x * , f ( t ) ) is differentiable, and by

=Sb$ ( x ’ , f ( t ) ) dt Lb( x * , f ’ ( O ) dr.

( x * , f ( b ) -f(a))

=

Thus if n* is a linear functional of norm 1 such that ( x * , f ( b ) -f(u)) Ilf(b)-f(a)ll

<Sbllf’(‘)ll dr

=

Ilf(b) - f(a)ll, we find

Ub) - S ( 4 ,

and consequently (2.1.21) holds.

2.1 D

Multilinear operators

The relations between boundedness, continuity, and differentiability are well exemplified by means of multilinear operators. An operator f E M ( X , Y) is called multilinear if X = IIy=’=,Xiand for each element x = (xl,x2, . . . , x N ) and each integer k = 1,2, . . . , N , f(xl, x2,. . . , x N ) is linear in xk with all other variables held fixed. Clearly every multilinear map is Gateaux differentiable. However, not every multilinear map is Frechet differentiable, and in fact we have (2.1.22) Let X = IIr.I Xi and Y be Banach spaces. Then iff multilinear, the following facts are equivalent:

E M(X,

Y) is

(i) f is continuous at each point x E X; (ii) f is continuous at x = 0; (iii) f i s bounded and there is an absolute constant K 2 0 such that (2-1.23) f(X,’ x2, * * * XN) G ~lIX11lX,II~2l1X, * * * IIxNlIX,; (iv) f is Frechet differentiable. 9

71


P m t : We show (ii)*(iii) and (iii)+(iv) since the other implications are obvious. (ii)*(iii): If f ( x ) is continuous at x = 0, there is some ball q, = ( x I llxll < 6) such that IIf(x)ll < I . Sincef(Kx) = K'"f(x),fis bounded on every sphere with center x = 0, so thatfis a bounded operator. Thus setting lllflll = sup Ilf(x)ll over the unit ball Z,, lllflll < 00. Without loss of generality, we may suppose that IIx(J= sup, Ilxillx,. With this choice of norm on X we demonstrate (2.1.23) with K = Illflll. The inequality is obvious if any xi = 0, so that we may suppose xi f 0 ( i = 1, . . . , N). In that case, setting y, = x , / ~ / x (, i~=~ 1, . . . , A'), SUP Ilf(yI,y2,. . . ,yN)ll G Illflll, and (2.1.23) follows with K = Illflll. (iii)*(iv): A simple computation shows that the Gateaux derivative a t x = ( x , , x2, . . . , x N ) is given by the formula

(t)

d'(x, h ) = x f ( x , ,

. . . , xi-

hj, x,+

I,

I,

. . . , xN),

J

where h = ( h l , h,, . . . , hN). Clearly, df(x, h) is linear in h; and so to establish (iv) we apply (2.1.23) after demonstrating that (a) d f ( x , h) G cllhll, and (b)f'(x)h = d'(x, h). If we suppose that llxill < M ( i = 1, . . . , N), (a) follows immediately from (t) by virtue of (iii). Thus, defining df(x, h) =J'(x)h, we find by (2.1.23) that after a short computation,

IlfG + h ) -f(x)

-f'(x)hll

=

4llhll).

Thus, f is Frechet differentiable.

Notation: The set of continuous multilinear operators f: X I x X , X * * * X X, -+ Y will be denoted by L ( X l ,X,, . . . , X,; Y); and we observe that by (2.1.23) and with the norm

lllflll = sup{l\f(xl~

' *

*

9

xN)ll/llxlll

IlxZll

*

L ( X , , . . . , X,; Y) is a Banach space. If XI = X , = write this space as L N ( X ,Y). A useful result in this connection is

*

*

-

I1xNII)'

.

= X , = X, we

(2.1.24) Lemma The Banach spaces L ( X , , X , ; Y) and L(Xl; L(X,, Y)) are identical up to a linear isometry. Proot We shall define two bounded linear mappings l1 and 3, that are inverses for each other with 11:L(X,,X,; Y)+L(X,;L(X2,Y)).To define 11,letf(x,, x,) E L ( X l , X,;Y)and regard x I as a parameter, so thatf(x,, x,) is a bounded linear mapping g E L(X,, Y), and set ll(f)= g. Clearly, 3, is linear, bounded, and of norm < 1 when regarded as a mapping of L ( X , , X,; Y)+L(X,; L(X,. Y)). To define lz, let g E L(Xl;L(X2,Y)). Then g ( x I ) x 2 =f(xI, x,) is a bilinear mapping of x l , x2 with values in Y. Letting lz(g)=f, the desired result follows since l2= 3;' maps L(X,;L(x,, Y))+L(X,,X,; Y ) .

A multilinear form f ( h , , h,, . . . , h,) E L,(X, Y) is called symmetric if it is invariant under all permutations of the indices (1. 2, . . . , N), a(1, 2, . . , n). Furthermore, with any multilinear form f(h,, h,, . . . , h N ) E L,(X, Y) we can associate a symmetric multilinear form E L,(X, Y):

.

(2.1.25)

1 f(hl,, Symf(h,, . . . , h N ) = N ! u ( i , , .. ., iN)

hi2, *

* * 9

hiN)

such that Sym f = f if and only if f is symmetric. In addition, for any symmetric form f ( h l , . . . , h,) E L , ( X , Y), the polar form f ( h )

12


f(xy=

. . . , h ) has the property that ,$hi) can be expanded by the multinomial theorem, so that we can find f ( h , , . . . , h,) by polarization according to the identity

= f ( h , h,

Thus if the polar forms of two symmetric multilinear forms f ( h , , . . . , h N ) and g ( h , , . . . , h,) are equal, then the equality of the multilinear forms, f(h,, . . * hN)=g(h,, . . . h N ) , follows. 3

9

2.1E

Higher derivatives

A (Frechet) differentiable operator j E M ( U , Y ) is twice (Frechet) differentiable at x iff’: X + L ( X , Y ) is differentiable (in the Frechet sense) at X . The derivative off’(x),f”(x)E t ( X , L ( X , Y ) ) = L,(X, Y ) by (2.1.24). Furthermore f E C2( U , Y ) if (a) f is twice (Frechet) differentiablefor each x E U , and (b) f ” ( x ) : U + L ( X , X ; Y ) is continuous. I f f is twice Frechet differentiable, the second derivative f ” ( x ) is unique; and we shall show that f ” ( x ) is symmetric, i.e., f ” ( x ) ( h , ,h2)=f”(x)(h2, A,). Continuing by induction, we can define Nth-order derivatives as N-linear forms as follows: f is N-times differentiable at x if the operatorf(N-’)(x) is differentiable (in the Frechet sense) at x. The derivative of f(N-l)(x),f “ ) ( x ) E L ( X , L ( X N - ’ ,Y ) ) = L , ( X , Y ) and can thus be regarded as a multilinear operator of order N . In addition, f E C N (U , Y ) if (i) for each x E U c X , f is N-times differentiable in the above sense, and (ii) as a function of x, f ( N ) ( x ) : U + L N ( X , Y ) is continuous. Again if it exists, f ” ) ( x ) is unique, and we shall show that f ” ) ( x ) is a symmetric N-linear form. Clearly, by substituting the notion of Gateaux differentiability into the definitions above, higher order Gateaux derivatives d N f ( x ,h ) can be defined. Essentially d d f ( x + th,, 4) d2f(x, h,, h2) = d ( d f ( x , h , ) , 4)= =- az

at, at, f ( x

1 .

+ t,h, + t2h2)

II= 1,=O

In the same way,

d N f ( x ,h,, h2, . . . , h N )= d [ d N - t f ( x ,h , , . . . , h N - , ) ? h N ] =

d [dN-tf(x dt,

+ t,h,,

h , , h,,

. . . ,AN-,

2.1

73

ELEMENTARY CALCULUS

Since the operators D i= a / a t , and Dj= a / a t , are commutative, d N f ( x ,h,, . . . , h , ) is symmetric if it exists. The relationship between these higher order derivatives is readily established by the following extension of (2.1.13). (2.1.27) Theorem (i) I f f is N times Frechet differentiable in a neighborhood U of x and f N ( x ) ( h l ,. . . , hN) denotes the Nth Frechet derivative, then f is N-times Gateaux differentiable and d N f ( x ,h , , h,, . . . , hN) = f N ( x ) ( h , ,h,, . . . , h,).

(ii) Conversely, if the Nth Gateaux derivative d N f ( x ,h , , . . . , h,) off exists in a neighborhood U of x, dNf(.x, h , , . . . , A N ) E L,(X, Y ) , and as a function of x, d N f ( x ,h , , . . . , hN) is continuous from U to L,(X, Y), then f is N-times Frechet differentiable and the two derivatives are equal at x. Proof: (i): Since f(x) is k-times Frechet differentiable ( k a multilinear operator A ( h , , . . . , hk) E L , ( X , Y ) such that

(*I

.

lIfk--(x + h,)(h,, . . . h,-J

-

fk-'(x)(h,,

< N ) , there is

. . . , A,-,)

- A ( h , , . . . hk)ll = O ( l l h k l l ) uniformly on bounded sets of ( h , . . . . h k - l ) . We proceed by induction: for k = 1, (i) is true by virtue of (2.1.13). Assuming the truth of (i) for n = k - 1, we prove (i) for n = k. Indeed we show dkf(x, h , , . . . , h,) = A(h,, . . . , h,). By the induction hypothesis 7

f"-'(x

and

+ h,)(h,, .

,

. , h,- ,) = d f ( x + h,, h,, h,,

f k - ' ( x ) ( h , , . . . , A,-,)

=

. . . , h,-,)

df(x, h,, . . . ,hk-1).

Thus (*) implies that as t +O, Ilt-'{dk-tf(x

+ rh,,

h,, h,,

. . . , A,-,)

. . . , h , & J } - A ( h , . . . . , hk)II = o(l), is k-times Gateaux differentiable and dkf(x, h , , . . . , hk) - d k - I f ( x ,h,,

so that f ( x )

. . . , h,), which proves (i) for n = k . This completes the induction argument, a n d thus f ( x ) possesses N Gateaux derivatives a n d dNf(x, h,, . . . , h,) = f N ( x ) ( h , , . . . , A N) . (ii): Again we proceed by induction. For n = I , (ii) is true by (2.1.13). Assuming the truth of (ii) for n = k - .1, we prove (ii) for n = k. We show that (*) holds with A ( h , , . . . h,) = dkf(x, h,, . . :, h,). By the induction hypothesis = A(h,,

.

f"-'(x

+ h,:)(h,, . . . , h k - 1 ) - f " l ( x ) ( h , , . . . , h k - 1 ) $ d k - t f ( x + sh,, h , , . . . , h,-J d;,

=I'

74


so that the right-hand side of (*) can be rewritten as

IIJ

1

0

d y ( x + shk, h , , *

-

>

hk) ds - dkf(X,

.

9

hk)ll-

By the mean value theorem (2.1.19), this last expression is IIdkf(X + t h k , h,, * * * hk) - d k f ( X , A,, * . * hk)ll (by hypothesis of the theorem). = o(llhkll) This completes our induction argument and proves (ii) for k = N. 9

9

(2.1.28) Corollary Under the hypotheses of Theorem (2.1.27) the Frechet derivatives f k ( x ) ( h I ., . . , hk) are symmetric for k = 2, 3, . . . , N. (2.1.29) Corollary Supposef E M ( U, Y), where U is an open subset of X,and that the line segment [ x , x h] c U , and also

+

(2.1.30)

IIf(x + h ) - f ( x ) -

X" 51 ai(x)h'II = O(IIhIIn),

i= 1

where a,(x)hi = a,(x, h, . . . , h) (the h repeated i times) are symmetric multilinear operators in L , ( X , Y) ( i = 1, . . . , n) which are continuous in x from V + L , ( X , Y ) . Then f E C " ( U , Y) and a,(x, h,, . . . , hi) = f i ( x ) (A,, * * * hi). 9

Proof: We show that if (2.1.30) holds then uj(x, h, h, . . . , h ) = (dJ/dt')f(x

+ th)l,,,

( j = 1,

. . . , n).

Indeed, lety* E Y* be arbitrary, then g(f) = ( y * , f ( x + th)) E C"(0, 1). By the one-dimensional Taylor theorem and (2.1.30),

By the uniqueness of Taylor series for real-valued functions,

Consequently, ak(x)h

=

dk

7 f(x + th)l,=,. df

Thus, for k = 2, 3, . . . , n , a k ( x , h , , . . . , hk) is symmetric. Hence u k ( x , h,, . . . , hk) = dkf(x, h,, . . . , hk) and (2.1.27) implies that ak(x, h,? * . * hk) = f k ( x ) ( h , , h,, 9

s

.

7

hk)

( k = 1, 2,

.

)a

Consequently, f E C"(U , Y ) . (2.1.31) Taylor's Theorem (weak form) If f is ( N - 1)-times FrCchet

2.1

ELEMENTARY CALCULUS

75

differentiable in a neighborhood U of x and f N ( x ) exists, then

+ h ) -f(x)

1

- f ' ( x ) h - . * . - - f ' " ' ( ~ ) h ' " I ( = ~(llhll"). n! Proof: (Cartan) For N = 1, the result is the definition of Frechet differentiation. Proceeding by induction, we assume the truth of the theorem for k = n - 1, and establish its validity for k = n. Consider the function I g ( h ) = f ( x + h ) - f ( x ) - f ' ( x ) h - . * - -f'"'(x)h", n! and calculate g'(h) by applying formula ( t ) of (2.1.27) for the derivatives of multilinear operators. Thus, 1 f"(x)h"-', g'(h) = f ' ( x + h ) - f ' ( x ) - * . . ( n - l)! (2.1.32)

Ilf(x

and by the induction hypothesis, 11 g'(h)ll = o(llhll"-'). Applying the mean value theorem (2.1.19), 11 g ( h ) - g(O)((= ~(llhll"). Since g(0) = 0, (Ig(h)l( = o(llhll"). Thus the induction argument is complete, and (2.1.32) is proven. (2.1.33) Taylor's Theorem Suppose f E C " + ' ( U , Y ) and the line segment [x, x + h ] c U , then (2.1.34)

f(x

+ h) = f ( x ) +f'(x)h + +f"(X)h* + . . . +

1 7 f " ( x ) h " + R,+'(X, h ) , n.

where

Proof: Let y* E Y*, then we apply Taylor's theorem for real-valued functions to the C"+' function g ( t ) = ( y * , f ( x + th)) so that for 0 < t < 1,

where

+

Now g(k)(t)= ( , V * , ~ ( ~ ) ( X t h ) h k ) so that gk ( 0 ) = ( ~ * , f ( ~ ) ( x ) hThus, ~). setting t = 1 and applying (2.1.35),

76


where 1

(1 - s)" ( y * , f " " ( x

+ sh)h"+') ds.

Thus by the Hahn-Banach theorem, (1.3.8) follows. 2.2 Specific Nonlinear Operators

In this section, we shall point out some concrete nonlinear operators that will be used in the sequel and discuss the representation of these operators in terms of mappings between Banach spaces. Suppose fl is a bounded domain in RN,and X and Y denote Banach spaces of functions defined over a.Then we begin by considering the following classes: 2.2A

Composition operators

Let f ( x ,y ) be a real-valued function defined on D X R' such that f ( x , y ) is continuous in y for almost all x, and for ally, f ( x , y ) is measurable with respect to x. Here i2 is a bounded domain in R N . In this case, f ( x , y ) is said to satisfy the CarathCodory continuiy conditions. Then we consider the nonlinear composition operator f ( u ( x ) ) defined by setting

-

j

( u ( x ) ) = f(x, u ( x ) )

(for u ( x ) Lebesgue measurable).

Clearly,j(u(x)) is Lebesgue measurable; and in fact, it can be shown easily * that f ( u ( x ) ) is continuous with respect to convergence in measure. Moreover, when regarded as a mapping of the Banach space C(a) into itself, the mapping f ( u ( x ) ) is also con.t.inuousand bounded. A deeper result concerning j that will be used frequently in the sequel concerns the -+ LP2(a).In particuproperties of j when regarded as a mapping of Lpl(fl) lar, we state:

(2.2.1) The following statements concerning the operator y ( u ( x ) ) are equivalent under the. growth condition' (*I If(x,y)l < a + f l ( y ( p J p 2 for some constants a,p 0: (i) j ( u ( x > ) maps L,,(Q) into LJD), 1 >is a continuous mapping of LJQ) into L ~ J Q ) . (iii) J ( u ( x ) ) is a bounded mapping of LJM) into Lp2(i2).

'

Actually the growth condition (*) is a consequence of (i). (See Note A at the end of the chapter.)

2.2 SPECIFIC NONLINEAR OPERATORS

77

The proof of these results follows from straightforward measure theoretic considerations and will be sketched in Note A at the end of this chapter. An easy induction argument shows that under suitable Caratheodory continuity conditions the multivariable composition operator f ( u l , u2, . . . , u k ) = f ( x , ul. . . . , uk) regarded as a mapping from T , ~ = ~ L ~ , ( Q ) - t L p ( 0 ) is continuous and bounded if and only if the function f ( x , y l , . . . , y k ) satisfies the growth condition

where the numbers ci are constants. As a simple application of (2.2.1), let us complete the proof of (1.5.7). In the special case proven there, it was assumed thatf(x, u ) = k ( l + u " ) with u < ( N + 2 m ) / ( N - 2m). Now, a close examination of the proof shows that the only way this explicit form was used consisted in guaranteeing the Lp boundedness forf(x, u ) for u in various Lp classes. Consequently, the u)l < k ( 1 + IuI") is all that is needed to result (2.2.1) shows that the growth condition preserve these Lp boundedness properties provided f ( x , u ) is Lipschitz continuous. Thus, a repetition of the proof given after the statement of (1.5.7) suffices for the general case.

,.(fI

2.28

Differential operators

A general differential operator A of order m defined over a domain

G

c [ w is~ written

(2.2.3)

A u = f(x, u, Du,

. . . , Dmu).

Usually, A is called an ordinary differential operator if N = 1, and a partial differential operator if N > 1. The operator A is quasilinear if (2.2.4)

A ( u , u ) = f ( x , u , Du,

. . . , D m - ' u , Dmu)

is a linear function of u when u is held fixed. A quasilinear operator A is semilinear if A ( u , u) = A ( u , 0 ) + A ( 0 , u), where A ( 0 , u) is a linear function of u and is independent of u. A differential operator Au is said to be written in divergence form if (2.2.3) can be written Au

2

= la/.

D"{A,(x, Dpu)}.

I PI c m

Clearly such operators are quasilinear. Operators in divergence form arise naturally since operators of this type are generally the Euler-Lagrange equations of some energy functionals of the Form Z(x, u , Du, . . . , Dmu) (cf. Section 1.1C). A classification of general linear differential operators extends immediately to a classification of large classes of nonlinear differential

78


operators. This can be carried out as follows: (i) If f E C ' and A u = f ( x , u, , . . , Dmu), one associates with A its first variation at u , namely A'(u)v =

x

fa(x, u,.

,

. , Dmu) D a v ,

where

la1 < m

fa

af =ata

*

Then one defines the ype of A at u by the type of the linear operator A'(u). (ii) If the operator A is quasilinear, this procedure implies that the

type of A is the type of the linear operator A ( 0 , v), provided that this type does not depend on a perturbation by lower order terms. Thus, for example, nonlinear elliptic differential operators can be defined in analogy with linear elliptic differential operators. In Chapter 1, a linear differential operator of order m, L =&al4 y ) )

4

9

where C ( x , y ) is Green's function for (A, a) relative to the null Dirichlet boundary condition. Thus, as a function of x, G(x, y ) = 0 for x E a Q and

where for fixed x, p ( x , y ) is a harmonic function of y . The operator A u defined by (2.2.8) when considered as a map of Lp(8)+ L,(O) can often be factored in the form A u = Lf(u),where f is the composition mapping from Lp(Q)+ L,(Q) for some r and L is the linear integral operator

W x ) = JQK ( x , y ) u ( y )4 regarded as a mapping from L,(Q).+L,(Q). Clearly, a sufficient condition

80


for L to define such a bounded linear operator is that where t = max s ( 9

5).

Another interesting example is the integral operator associated with solutions of the Neumann problem for a domain Q c RN,

Au = 0, Apart from an arbitrary constant, any solution of such an equation can be written in the form

where N ( x , y ) is Green's function for the Neumann problem for A, and so has a representation analogous to that of G ( x , y ) given above. 2.20

Representations of differential operators

There are several distinct approaches that can be used to represent general differential operators by abstract nonlinear mappings between Banach spaces. The methods that will be useful in the sequel are summarized as follows. (More detailed discussions are given later.) (I) Direct composition representations If Wu = f ( x , Dpu), I < m, is a differential operator of order m defined on some domain Q c RN and f = f ( x , 5) is smooth in (x, t),say of class C"., then we can consider the composition operator 62 ( u ) = f ( x , Dpu), 1 PI Q m, defined for u on . a Holder space of functions C""(Q). Thus for u E C s 3 " ( Q ) ,@ ( u ) E Cs-">*(Q), provided of course that s 2 m. Such a mapping is clearly continuous and bounded from C"."(Q) + C s - m * a ( Q )In . fact the Frechet derivative of Q ( u ) at uo E Cs,"(Q), A'(uo)v, is easily computed to be

(2.2.9)

A'(uo)v =

2

fa(.,

uo, Dpuo) D a v ,

1.1 < m

where fa = af/a(,. The expression on the right of (2.2.9) is just the first variation of W ( u ) at uo. In the same way & ( u ) can be regarded as a provided that the formal derivatives mapping from W ,,(a) + W,-,,,(Q) Dyf(x, . . . , D p u ) E L,(Q), J y I < s - m , with u E WS,,(Q).This last restriction can be verified by placing growth restrictions on f and its derivatives similar to those of (2.2.7). A useful example in studying the boundedness of A is the following:

81


(2.2.10) Supposef(x, 5) is a C“ function defined on a domain a X R, all of whose derivatives are bounded. Then for an integer m sufficiently large, and any u E Wm,p(Q), IIf(x, U)IIm,p

‘

const. ( 1 +

IIuIIm,p}.

Proof: For simplicity we consider the case in which f is independent of x . Then, by the chain rule one computes that formally, for u E Cnm(Q), D k f ( u )=

5

J=

To estimate bounded,

(

~,~fO)(u) I

D h )

( k = 1, 2, . . . ).

2PJ=k

I l f ( ~ ) l l ~ , ~ , it suffices to estimate IIDmf(u)llo,p.Since by hypothesis thefO(u) are I l ~ k f ( ~ ) I I % , p6 const.11

n

@#.p.

Z&=k

By Holder’s inequality with p,p, = m,

In the same way we can show that llf(x, u , D K . * . D y u ) l l m - y , p Const. { 1 + provided m is sufficiently large.

(2.2.11)

3

II~llm,p),

(ii) Operators defined by Schauder Inversion An approach used to great effect since the fundamental studies of J. Schauder is based on the inversion of differential operators (possibly supplemented by appropriate boundary conditions). The basic idea used to define abstract nonlinear mappings associated with boundary value problems for quasilinear differential operator equations A u = g consists in writing A ( u ) = A ( u , u ) in such a way that:

(1) for fixed elements (0, g) in carefully chosen Banach spaces ( X , Y ) , the linear equation A ( u , u ) = g has one and only one solution u = Tg(u)in X ; and (2) A ( u , u ) depends continuously on u for fixed u E X .

Then the operator Tg is well defined, so that Tg E M ( X , X ) and the fixed points of Tg coincide with the solutions of A u = g . In order to assert the existence of such fixed points of Tgr it will be important to establish the continuity and boundedness of Tg E M ( X , X).

82


One generally establishes these facts by deriving a priori estimates for solutions u of the equation A ( u , u ) = g with u and g fixed and llullx < R of the form (2.2.12) llullx G c(R)II gll Y ) where c ( R ) is a finite positive constant independent of u and u but possibly depending on R . For example, since the differential operator A ( u ) of order m is quasilinear, one can assume, to establish estimates, that the operator A ( u , u ) is linear in u with u used to denote only derivatives of order m, and u derivatives of lower order. The continuity and boundedness of Tg then follow in many instances from the following: (2.2.13) If A ( u ) is quasilinear and the linear operator A(u, u), defined by fixing u and letting u denote the mth-order derivatives in A ( u ) , admits an estimate of the form (2.2.12), then the operator Tg (so defined) is continuous and bounded as a mapping of X into itself. Proof: Once (2.2.12) is assumed, the boundedness of Tg is immediate since sup IIT,(u)ll < 00 over llullx < R. The continuity of Tg is established as follows. Suppose Tgu = u and Tgij= ii, then A (u, u ) = g and A (6, a) = g. Consequently, since A is linear in its second argument, we find A(G,~-u)=A(G,u)-A(~,~)=A(ij,u)-A(u,u).

By virtue of the estimate (2.2.12) and the above IITgG - Tgull = llE - UII G c(R)lIA(ij, .) - A ( u , u)ll, where R = max(llull, IlVll). Thus as G+u, since A ( u , u ) is continuous in u for fixed u, T,G+ Tgu,as required. For the study of elliptic boundary value problems of order 2m, typical pairs (X,Y) for the Schauder inversion procedure are the Holder spaces (C2"~'(51),Co.'(51)) (0 < a < 1) and the Sobolev spaces ( Wzm,p(Sl)rLp(51))(1 < p < m). Indeed, for such pairs the estimates (1.4.25)-(1.4.28) can be used to verify (2.2.12). Actually the estimates (1.4.25)-(1.4.28) prove that the operator Tg is also compact (see (2.4.7)),a fact that will be important later. For semilinear elliptic equations, the operator Tg can be exhibited quite explicitly by Green's functions. Thus, for example, if G(x, y ) is Green's function for the linear differential operator Lu = Z,,,,,,,a,(x) D'u defined on 51 c RN and subject to the Dirichlet boundary conditions D%J,, = 0, JaJ< m - 1, then the operator Tg described above for the nonlinear system L u + f ( x , u , D u ,..., DBu)=O D%J,, - 0

in51, I @ l < r n - l , for la1 < m - 1

coincides with the integral operator T ~ x= ~ ) ( x . y l f ( yu,,

. . . ,DBu(y))4.

(111) Operators defined by duality For differential operators A , defined on a domain 52 c R", of divergence form

.

( - l)'%X4,(x, u, . * , D%),

Au =

I4 < m

83


an especially effective abstract integral representative (2, based on the reflexivity of the Sobolev spaces can often be defined. Suppose that for u E Cp(S2) and + E CT(Q), we set

2

F ( u , +) =

J A , ( x , u , . . . , D"u) D"+,

lal<m

Q

and by virtue of the inequalities of Section 1.4 we can prove the existence of a bounded function g ( r ) such that for m a positive integer and I 0, f can be extended to a compact operator f E K ( X , Y ) in such a way that for x E x , d ( f ( x ) , m f ( U ) ) Q 8. Proof: By Theorem (2.4.2), f ( x ) = x r = o _ f , , ( x ) By . Tietze’s extension theorem, each j l ( x )can be extended to an f,,: X + Ys, with preservation of norm. Hence is a compact map. Now consider the mapping f ( x ) For x E X ,

=xr=ofn(x).

x+ W

d(fo(x), i35f(W) G l l f o - fll

=

€9

n= 1

also d( CO fo( U ) , i35 f( U ) ) < c since for x E C6 fo( U ) , P

r,fo(.x,)

x= I=

so

with

‘J =

1

and x, E U ,

1

z

z4

Ilfo(x,> - f(x,)ll Q Then choosing 3c = 8, f is the desired extension since Ilx -

d ( j (x),

‘,f(X,>II

Q

f(U)) G

llf

<x 03

n= I

(x) - fo(x)II

+ d(fo(x)* i3 f( U ) )

E

-++E=2c<S 2“

We now make some remarks concerning the relations among compactness of an operator and some other basic notions already introduced. (2.4.5) Suppose f E M ( U , Y ) is completely continuous, where U is a subset of a reflexive Banach space X . Then f is compact. Proof: Suppose { x,,} is a bounded sequence of U . Then by the reflexivity of X , {x,,} has a weakly convergent subsequence {x,,,}. The complete continuity off implies that { f ( x , ) } is convergent in the norm topology of Y . Thus f is compact.

(2.4.6) Suppose U is an open subset of X and f E K ( U , Y ) is Frechet differentiable in U . Then the Frechet derivative f ’ ( x o ) E L ( X , Y ) is a compact linear operator for fixed xo E U . Proof: Suppose j ’ ( x o ) were not compact. Then, if u , denotes the unit sphere in X,fl(xo)(a,) does not have compact closure in Y . Hence there is a sequence { h i } with llh,ll = 1 and a number E > 0 such that Ilf’(xo){hi -

92


h,}ll > E ( i # j ) . On the other hand, for P Frechet differentiability off at xo implies,

-f

Ilf(x0 + Ph,) -

Ilf(x0

b o + Ph,)I/

+ Phi) - f(.o)

- Ilf(x0 + PhJ

2

PE

-

> 0 and sufficiently small, the

P Ilf’(xo)h, - f’(xo)h,II - Pf’(xo)h,ll

-f(xo) - Pfl(xo)h,II

4 PI).

Since c is independent of P, this last inequality implies that { f(xo + Ph,)} has no convergent subsequence, which is the desired contradiction. 2.4C Compact differential operators

Heuristically speaking, operators that possess some definite “smoothing property” are generally compact. As a very simple example we consider the operator T f ( x ) = J&f(s) ds defined on C [0, 11. Clearly Tf is differentiable on (0,l), and so T possesses a smoothing property in the sense that T maps continuous functions$into differentiable ones. On the other hand, T is compact by virtue of the Arzela-Ascoli theorem (1.3.13). This argument can be extended to the more general abstractly defined operators of Section 2.2D by a careful inspection of the estimates of Section 1.4, as follows: (i) Consider, for example, the class of abstract operators defined for quasilinear elliptic differential operators by means of the Schauder inversion method of Section 2.2. In this connection, we prove the following abstract result.

(2.4.7) Lemma Suppose the operator A ( u , u ) is linear in u and continuous in u for fixed u E Z and maps X x Z-+ Y , where Z is a linear subspace of X compactly imbedded in X . Then, if the linear equation A ( v , u ) = g has one and only one solution u = T’u for (Ilullx < R ) satisfying the a priori estimate (2.4.8) Il4lz c(R)II gll Y7 where the positive constant c ( R ) is independent of u, the mapping Tg:X+X is compact. Proof: The continuity of Tg follows as in (2.2.13) since Z c X . To demonstrate the compactness of T,(u) for any bounded set u in X, suppose { u , , } is any sequence of u. Then if u,, = T,(u,,), the estimate (2.4.8) shows that

I1 T’(%)llZ

c(u)ll

gll Y )

2.5 GRADIENT MAPPINGS

93

where c(u) is a constant depending only on u. Since Z is compactly imbedded in X , any bounded set in Z is compact in X . Thus T,(l;,,) has a convergent subsequence in X . Consequently, Tg is a compact mapping. For elliptic differential operators defined on bounded domains s2 with normal homogeneous boundary conditions, the estimates ( I .4.26) and (1.4.28) yield (2.4.8) and the compactness necessary to apply Lemma (2.4.7) to the pairs

c"(a))

( 2 ,X ) = ( C 2 m - a ( 8 ) ,

or

( z ,X ) = ( W2m,p(s2),I#))

forO 0 holds for each x, y E X .

+

+

Proot: (i): By virtue of (2.5.2(iii)), if one sets @ ( x ) = IA(f(sx), x) ds, then ( d / d r ) @ ( x ch)l,,, = (f(x),h). Since this equation is true for each h E X,W ( x ) = f ( x ) ; and since

97

2.5 GRADIENT MAPPINGS

Cg(0) = 0, the uniqueness of the Gateaux derivative implies F ( x ) = @(x). (ii): I f f is a gradient mapping and x,, -+ x weakly in X, then by (2.5.2(iii)), F(x,) = - x,j(xn(s))) ds, where xn(s) = x + s(x, - x). Thus: writing

-

F(x)

(x, - x, f"(S)) = (x, - x, f(Xn(.)) - f(x)) + (x, - x, f(x)), we note that since {llx,ll) is uniformly bounded

lim { F ( x , ) - F ( x ) ) =

n+m

J i 'i ( x% n - x , f , ( s ) ) ds = 0.

Indeed, f is completely continuous, so the limit on the right-hand side is zero because f ( x , ( s ) ) - + f ( x ) strongly for each s E [0, 11 (while the integrand on the right-hand side is uniformly bounded). (iii): Set @(x) = ( I / ( n + I))(f(x), x), then using the presumption that @(x,, . . . , x,+ I) = (f(xI, . . . , xn), x,+ ,) is symmetric, we find that for any h E X , (d/dc)@(x + ch)J,=, = ( f ( x ) , h). Hence @(x) is the antiderivative of f ( x ) . (iv): First we suppose that ( f ( x ) -f(y), x - y ) > 0 for each x , y E X (i.e., that f is a monotone operator) and establish the convexity of F ( x ) . Indeed, without loss of generality we may suppose that F(0) = 0, so that by (i) above, for any I E [0, I],

(this last inequality following from the monotonicity of 1).By (2.5.2(iii)), the right-hand side of the above inequality can be written as I ( F ( x ) - F ( y ) ) . Thus F ( f x + (1 - r)y) G r F ( x ) + (1 - t ) F ( y ) .We now prove the converse; namely, if F ( x ) is convex, then ( f ( x ) - f(y), x - y ) > 0 for each x , y E X. To this end, we first show that the convexity of F ( x ) implies for each x , y E X . F ( x ) + ( f ( x ) , y - x) < F ( y ) The convexity of F ( x ) implies that for each x , y E X and s E [0, I],

(*I

sF(x)

+ F ( x + " ( y - x))

-

F(x)

< sF(y).

Dividing by s and letting s+O, we find that (*) results. Next we apply (*), interchanging the roles of x a n d y to find

(**I Adding

F ( y ) + (f(y), x (8)

y) G F ( x ) .

and (**). we find ( f ( x ) - f(y). x

2.5C

-y

) > 0, as required.

Specific gradient mappings

Generally speaking, if a differential operator A is the Euler-Lagrange derivative of a functional (in the sense described in Section l.lC), A can be represented abstractly as a gradient mapping (h' . In order that the mapping @ be defined in Sobolev spaces, the terms of A must satisfy cerlain growth conditions. As an interesting nontrivial example, we consider the partial differential equations defining the von Karman equations given in Section 1.3B.It is known that generally the deformation equations of elasticity are derived as Euler-Lagrange equations, so it is plausible to expect that the

98


associated operator equations involve only gradient operators. In fact we now prove that our calculus guarantees this fact. (2.5.7) The weak solutions of the equations (1.1.12) are in one-to-one correspondence with the !olutions of the operator equation u + Cu = ALu in the Sobolev space W2,2(Q),where L is a self-adjoint mapping of W2,2(Q)into itself and Cu is a gradient mapping of W,,,(Q) into itself. Moreover, there is a symmetric bilinear mapping C(u, v ) of W2,2(Q)into itself, defined by (2.5.9') below, such that Lu = C(F,, u ) for some fixed element F, E W2,,(Q)and C(u, C(u, u)) = Cu. Proof: First we note that (without loss of generality) we may set c = 1 in the equations (l.l.l2), and defining F, as the solution of A2F= 0, DVF(,, = A$,, we may write a solution ( u , F) of (1.1.12) in the form ( u , f + AF,), so that the pair (u,f, satisfies the system:

- t [ u , u], A% = A[ Fo, u ] + [f, u ] , A'f=

(2.5.8)

D % = O a f = 0,

la[ G 1,

+

where [f, g] = (f,g, - fxygy)x (fxxgy - fxygx)y, Consequently, in accord with the definition of weak solution given in Section 1.5 and by choosing the inner product in k2, 2(Q) as (u.

u)2.2=

&xx~xx2u.v~xy+ uxxuyy)?

the weak solution (.,A of the system (2.5.8) can be written as any pair ( u , f, that satisfies the following two integral identities for all cp, 1) E C c ( Q ) :

+

, f = AFo f. Now we define the bilinear operator since $ [u, u] = ( U , U ~ , )-~ ( u , ~ u , ) ~where C(w, g) by means of rhe duality mefhod (cf. Section 2.2D(iii)) by setting

(2.5.9')

(C(w9 g). cp)

' j , { ( & p-ygyy~,)cpx + (gxywx - gx,~,)cpy)

for g, w, cp E H. The operator C is easily seen to satisfy the following properties: (i) (C(w, g), cp) is a symmetric function of g, w, cp (this follows by integration by parts); where K is an absolute constant. This follows (ii) (C(w, g), cp) c KJIg112,211wlll,411~lll,s, by Sobolev's imbedding theorem and Holder's inequality. Thus the system (2.5.9) can be written in the form (relative to the inner product in H) ( u , 1))

= (c(u,j ) ,v),

(f, cp)

= - (C(U, 4.cp).

Since 7,cp are arbitrary, we may write these as u = C(u,f)

+ AC(u, Fo),

f = - C(U, u).

Thus, setting C(u) = C(u, C(u, u)) and Lu = C(u, F,), we may rewrite these (2.5.10)

(a) u + Cu = ALu,

(b) f - - C ( u , u ) ,

where it is understood that any solution u of (a) determines f from (b) uniquely.

2.6 NONLINEAR FREDHOLM OPERATORS

99

Now the fact that C(u) defined above is a gradient mapping is now an easy consequence of the fact that the form ( C ( w , g), ‘p) is symmetric in w , g, and 9 . Indeed, in accord with (2.5.2) or (2.5.6), a short computation shows that if we set I ( u ) = ( C ( u ) , u), then d(I(u + €0)) /dcIc-, = (Cu, 0) for all u, 11 E H . For the same reason, the operator Lu = C(u, F,) is self-adjoint.

a

A simpler but nonetheless significant example concerns the semilinear operator A u = Au + f(x, u ) defined on a domain Q c RN. Such an operator into itself can always be represented as a gradient mapping of provided the function f ( x , u ) satisfies the smoothness and suitable growth cpnditions. Indeed, to use the duality method of Section 2.2, suppose f(u) = f ( x , u ) defines a bounded operator from @,,,(a) into Lp for p < ( N + 2)/(N - 2). Then the abstract operator d u defined implicitly by the formula ( & u , u) =

L{v~.

v v -j(x, u)u)

for

u E CT(Q)

can easily be verified to be a gradient mapping with antiderivative I(U) =

I a

{ f 1 V u12 - F ( x , u ) } dV,

where Fu(x, u ) = f ( x , u).

2.6 Nonlinear Fredholm Operators A smooth mappingf between Banach spaces X , Y can be studied by properties of its Frechet derivative f’(x). This approach was adopted in Section 2.3 for complex analytic mappings and for gradient mappings in (2.5). In the same direction, based on the results of Section 1.3F, we consider the following: 2.6A

Equivalent definitions

(2.6.1) Deflnltion Let X , Y be Banach spaces and U a connected open subset of X . A mapping f E C ’( U , Y ) is called a nonlinear Fredholm operator if the Frechet derivative of f , f ’ ( x ) is a linear Fredholm map E L ( X , Y ) for each x E U (see Section 1.3F). In this case, the index of f, ind f, is defined by setting ind f ( x ) = ind f ’ ( x ) = dim Ker f ’ ( x ) dim cokerf’(x) for x E U . (2.6.2) indf(x) is independent of x E U . Indeed, since f ’ ( x ) is continuous in x, ind f: U + 7 is continuous; and since U is connected, x E U implies indf(x) is constant. Thus, indf(x) is independent of x E U . (2.6.3) Examples of Fredholm maps and the computation of their indices are readily obtained.

100


(a) Any smooth map between finite-dimensional Banach spaces is a Fredholm map. (b) Any diffeomorphism between Banach spaces is a Fredholm map of index zero. (c) If f ( x ) is any Fredholm map and C(x) E C ’ ( U , Y ) is a compact operator, then f C is a Fredholm operator and ind(f C) = ind f. This result follows from (2.4.6). Indeed, ind(f+ C) = ind(f’ + C’) = ind(f’) since C’ is compact, and ind(f’) = indf.

+

+

(2.6.4) Theorem Let f E C’( U , Y ) , then for U an open subset of a Banach space Y , the following statements are equivalent: (i) f is a Fredholm operator. (ii) For each fixed x E U , the following inequalities hold for each y E Y: (2.6.5)

llyll

CiIIf’(x)YII + IYIo,

(2.6.6) llYll c,llf’*(x)Yll + IYIP where the constants C, and C, are independent of y and (yl0and ( y I I are compact seminorms defined on Y . Proof: By virtue of (1.3.37), the inequalities (2.6.5) and (2.6.6) taken together imply that for each x E U , f ’ ( x ) has closed range and that dim Kerf’, dim Kerf’* are finite. Conversely, (1.3.37) also implies that any linear Fredholm mapping f ’ ( x ) E L ( X , Y ) satisfies inequalities of the form (2.6.5) and (2.6.6) for each x E U . Thus (i) and (ii) are equivalent. 2.68

Basic properties

The theorems of Morse and Sard as mentioned in (1.6.1) have useful extensions for nonlinear Fredholm mappings. These extensions will be taken up in Chapter 3. As a first step in this direction, we define the notion of singular and regular points of differentiable operators. (2.6.7) Definitlon Let f E C ‘( U , Y ) , then x E U is a regular point for f if f ’ ( x ) is a surjective linear mapping in L ( X , Y ) . If x E U is not regular, x is called singular. Similarly, singular and regular values y off are defined by considering the sets f - ‘ ( y ) .I f f - ‘ ( y ) has a singular point, y is called a singular value, otherwise y is a regular value. In this connection, as in the finite-dimensional case, we prove (2.6.8) Theorem The singular points of a Fredholm operator f C ‘ ( X , Y ) are closed.

E

101

2.6 NONLINEAR FREDHOLM OPERATORS

).('fI

Proof: Let S = { x is not onto} and suppose x, E S is such that x,+E. By the continuity of the index of f, under small perturbations, index f ' ( x n ) = index f'(E) for n sufficiently large. Also by Theorem (1.3.38) of Chapter 1, if 11 B 11 is sufficiently small, and A is a Fredholm map, (2.6.2) implies that dim coker(A B ) < dim coker A . (2.6.9)

+

Hence, for n sufficiently large, (2.6.10)

dim cokerf'(K) 2 dim coker[f'(T)

+ ( f ' ( x , ) -f'(E))]

2 dim cokerf'(x,)

> I.

Consequently, f'(x) is not a surjective linear map, and

SO

x E S.

2.6C Differential Fredholm operators The class of nonlinear Fredholm operators arises very naturally in the study of differential systems since many differential operators (possibly supplemented by auxiliary boundary conditions) and their adjoints have only finite-dimensional subspaces of solutions. Now suppose we are given the nonlinear elliptic operator (2.6.1 I )

N ( u ) = F ( x , u , . . . , D%)

defined on a bounded domain Q c R N subject to the Dirichlet boundary conditions Dnulan = 0, (a(< m - I. Then we can consider N as a mapping of C2m,u(Q)+Co,a(Q).Provided the Frechet derivative the function F = F ( x , 6 I , . . . , t2")is a C' function of (6 I, . . . , t2"'), of N ( u ) at uo is easily computed to be (2.6.12)

N'(uo)u =

la1 < 2 m

F , ( x , uo, . . . , DZmuo)D"o.

By virtue of the Schauder estimates (1.4.27) and (1.4.28), there is a constant c possibly depending on uo such that (2.6.13)

II w II C'".

c { I1 " ( u o b

II co. + II4 co.-1.

Thus, I ~ U ~ ( ~isO a. ~compact seminorm defined on C2"'sa(Q).N'(uO) has closed range in Co9" and finite-dimensional kernel. To show that N is a nonlinear Fredholm operator one must verify that the system (2.6.14)

N ' ( u ~ ) u = ~ , Dnolan=O,

f~ C0.",

can be solved apart from a finite-dimensional subspace of Co.a(Q).To this end, we note that ( ~ ., . . , D2"uo) Dao maps C2m,a(51) onto Co."(Q) and is the operator L,(u) = Z I m , - 2 m F nuo, also one-to-one. On the other hand, L2(u) = 2 1 n 1 ~ 2 m - I Fuo, m (. x. ., , D2mUo)D"o is compact as a map from C2'"*"(Q) to C0*"(Q).Thus N'(uo) = L , + L,= L , ( I + L;IL,) can be factored as a homeomorphism acting on a compact perturbation of the identity. Applying the theory of compact operators to I + L;'L, we find that the system (2.6.14) is solvable provided f is orthogonal to a finite-dimensional subspace in Co*"(51). Thus N is a nonlinear Fredholm operator of index zero.

102


2.7 Proper Mappings 2.7A Equivalent definitions

An operatorf E C ( X , Y) is said to be proper if the inverse image of any compact set C in Y,f-'(C), is compact in X. The importance of this notion resides in the fact that the properness of an operator f restricts the "size" of the solution set Sp = ( x I x E X , f ( x ) = p) for any fixed p E Y. Thus, it is immediate that the only proper linear operators in L ( X , Y) are one-to-one and have closed range. More generally, we prove (2.7.1) Theorem Let f E C ( X , Y), then the following statements are equivalent : (i) f is proper. (ii) f is a closed mapping and the solution set Sp = ( x I x E X , f ( x ) = p) is compact for any fixed p E Y. (iii) If X and Y are finite dimensional, then f is coercive (in the sense + co). that IIf(x)ll+ 00 whenever I1xJ( Proof: (i)+(ii): Since any point p E Y is compact, the properness of f implies that Sp is compact. To prove that f is closed, let K be a closed subset of X and suppose y, = f ( x , ) +y, x, E K. Then, since the closure - of { y,,), { y,} , is compact, the properness off implies that u = f-'( { y o }) is compact. Consequently (after possibly passing to a subsequence), since x, E u, {x,) converges to a point Z.Since K is a closed set, 2 E K, and by the continuity off, f(X) = y . Now suppose that f is closed and for any p E Y, Sp is (ii)-(i): compact. Then, to show that f is proper, let C be a compact subset of Y and f - ' ( C ) = D.Suppose that D is covered with closed sets D, that have the finite intersection property. We show that n .Da # 0 , implying that D is compact. To this end, let ( a l ,. . . , ak)= /Ibe any subset of (a). Then Es = n: = I D , is closed and nonempty, so that f ( E p ) is closed and C = U Bf(Ep)'.Furthermore, the closed sets f ( E s ) have the finite intersection property since for any finite subset y E 2*,

nY f ( E p ) I f (n Y ~ g + )

0.

Therefore the compactness of C implies that 6 = n J ( E s ) # 0. Now let y E S and Dy = D n f-'(y) so that Dy # 0. By hypothesis, f-'(y) is compact, as is the set Dy= U ,{ D, n f-'(y)). Thus it suffices to show that (D,n f-'(y)) has the finite intersection property since then

n D,> n P a nf-'(v)} a a

# 0.

103

2.7 PROPER MAPPINGS

Finally, for any finite subset y E

n

=

{ a I ,.

. . , a,}

{ a } , since y

of

Ep)> j

n1 D ,n f - ' ( y ) = E, n f - ' ( y )

# 0.

;=

(iii)w(ii): Let X and Y be finite dimensional. Then the properness off implies that the inverse image of a bounded subset of Y is bounded in x, which is merely a restatement of the coerciveness off. Conversely, i f f is coercive and C is any compact subset of Y , thenf-'(C) is bounded and so is relatively compact in X .

For special classes of operators f E C ( X , Y ) acting between infinitedimensional Banach spaces, the coerciveness off implies the properness of f. More precisely, we prove the following criteria for properness. (2.7.2) Suppose f E C ( X , Y ) and IIf(x)ll+ co as llxll+ proper if either

00.

Then f is

(i) f is a compact perturbation of a proper mapping; or (ii) X is reflexive, and x,+x weakly in X with { f ( x , ) } strongly convergent, implies that x, + x strongly. Proof: (i): Let f ( x n ) = y,with yn +y in Y . Then, if f ( x ) = g ( x ) + C(x), where g is proper and C is compact, the coerciveness off implies that { x,} is bounded. Consequently, after possibly passing to a subsequence, { C(x,,)} is convergent. Thus, since the sequence g(x,) = yn - Cx, is convergent while g is proper, { x,} has a convergent subsequence { x,,} with limit X. The continuity off then implies that f(X) = y ; and so f is proper. (ii): If X is reflexive and f(xn)+y in Y , then the coerciveness of f implies that {x,,} is bounded. Hence (after possibly passing to a subsequence, once more), we may suppose x, + K weakly in X and hence, by hypothesis strongly, so that f(X) = y , so that once again f is proper.

2.76

Basic properties

A simple quantitative property of a proper mappingf E C ( X , Y ) is the following one expressing the stability of the solution set S,(n= { x I x E X , f ( x ) = p } under small perturbations in p or f.

(2.7.3) Theorem Let f E C ( X , Y ) be proper. Then: (i) for everyp E Y and every (2.7.4)

I l f ( x ) -pII

<S

E

> 0, there is a 6 > 0 such

implies

IIx - f - ' ( p ) l l

< E;

that

104


(ii) if g E C ( X , Y ) , then IIf(x) - g(x)ll Q 6 for all x E X implies that

W,,(f),S p ( g ) )Q E . Proof: It suffices to prove (i) since (ii) is an immediate consequence of (9. Thus suppose (i) is false. Then there exists an E > 0, a p E Y , and a sequence {x,} E X such that for all n

(2'7.5)

I I ~ ( X ~ ) - PQI I l / n

and

IIxn - ~ - ' ( P ) I I

€*

Since f is proper and f ( x , ) + p , by passing to a subsequence if necessary we may suppose that x, + x . Then since f E C ( X , Y ) , f ( x )= p and x ~ f - ' ( p ) But . this fact contradicts (2.7.5). In the same direction we prove (2.7.6) Let X and Y be Banach spaces and f E C ( X , Y ) . Suppose U and V are open subsets of X and Y , respectively, such that f maps U onto V , is locally invertible, and proper on U . Then the function c,, = the number of points in S,,(U) = { x I x E U , f ( x ) = p } is finite and constant in each component of f( U ) . (See Fig. 2.1 .) Proof: Clearly, the local invertibility and properness of f imply that f - ' ( p ) is discrete and compact. Consequently, cp is finite. The fact that c, is locally constant follows in the same way from Theorem (2.7.3), just obtained.

More generally, we now consider proper mappings that are not locally invertible. As in (2.6.7), iff E C ' ( X , Y ) we say that x is a singular point if f ' ( x ) is not locally invertible at x. Let the set of such singular points forf be called the singular set S . With the same notation and terminology as in (2.6.7), we then prove

FIG. 2.1 A typical decomposition of the range of a proper Fredholm mappingf by its For p E Oi the equationf(x) = p has exactly i singular values, into connected components 0,. solutions.

2.7

105

PROPER M A P P I N G S

(2.7.7) I f f E C ' ( X , Y ) is a proper Fredholm operator of index zero and S denotes the singular set o f f , then cy is constant on every (connected) component of Y - f(S). (More generally for proper operators of higher index the sets f - ' ( y ) are homeomorphic.) Proof: Clearly by (2.6.8) S is closed; and sincef is proper, (2.7.1) implies = X - J - ' ( f ( S ) ) and V = Y - f(S) are open subsets of X and Y , respectively. Now we can apply (2.7.6) to U and V . Clearly. f maps U into V . is proper, and is locally invertible on U . Thus the result follows since the components of V are arcwise connected.

f(S) is closed. Then U

2.7C Differential operators as proper mappings Finally, we investigate some criteria for the properness of abstract mappings associated with nonlinear differential operators. First, consider the concrete operator ~~

~

@u =

x (-

la1

I)'%,

(A&,

u,

. . . , D%)}

m

defined on a bounded domain Q c RN, and the abstract operator A : @y,p(Q)+ W - m ; q ( Qassociated ) with ti' by the formula (using the duality principle of Section 2.2D) (2.7.8)

( A u , 9)=

2

] A , ( x , u,

. . . , D*u) D U q ,

/al 0. This reduces to proving that the following limit can be taken under the integral sign:

(0)

The justification for taking the limit under the integral sign is based on Vitali's theorem on absolutely equicontinuous integrals and the given growth condition. (iii) f is bounded. This result follows from the continuity off at zero, and the continuity of Lebesgue measure. Although we shall not use this in the text, it can be shown that the growth condition (*) is a consequence of the fact thatfmaps Lp,(D) into Lp2(D). For this proof, the reader is referred to Krasnoselski (1964). B Real analytic operators

Our development of complex analytic operators was based on Hartog's theorem on separate analyticity. The analogous situation for real Banach spaces has not been extensively developed. A smooth mapping f defined on an open set D of a real Banach space X with range in another real Bmach space Y is called real analytic in D if it possesses Frechet derivatives of all orders at each point in D, and moreover f ( x ) can be expanded as a convergent power series in terms of these derivatives as in (2.3.3 (vi)). Certain results on real analytic operators can be deduced from analogous results on complex analytic operators. Indeed, each real Banach space can be imbedded isometrically into a complex Banach space X + iX in a canonical manner; and in fact bounded multilinear, symmetric: mappings of X into Y can be uniquely extended as multilinear symmetrlc bounded mappings of X + iX into Y + iY. Thus one shows that a real analytic map can be extended canonically to a complex analytic one. For these results, the reader is referred to Alexicwicz and Orlicz (1954). C The abstract Navler-Stokes operator

The Navier-Stokes equations (1.1.18)-(1.1.19) can be reformulated via the duality method of Section 2.2.D as an operator equation in a Hilbert space H. The condition div u = 0 allows one to restrict attention to solenoidal N-vectors and the Hilbert space H can be chosen to the space of solenoidial N-vectors w obtained by completing each component of u E CF(D) in the Sobolev space W,,@). If we consider the Navier-Stokes equations defined on a bounded domain 0 in RN, N = 2, 3, subject to homogeneous boundary conditions of Dirichlet type we find that these equations can be written in the form fA(w)

w

+ hNw = g

(A = Reynolds number),

108


(Note that the pressure term, appearing as a gradient, drops out of these equations and represents an extended forcing vector.) Using the duality method to justify this fact, given the fact that N is defined implicitly by the formula

It is easily proven that (i) The operator N so defined is a compact mapping of H into itself. (ii) For each y , the massociatedoperator f, is a proper mapping. (Note here that ( N w , w) = 0 for each w E H , so that the properness follows via (2.7.2).) (iii) fu(w) is Frechet differentiable and consequently f v ( w ) is a nonlinear Fredholm operator of index zero. (iv) For sufficiently small y , the solution of f v ( w ) = g is unique. As we shall see in Chapter 5 , for a large class of inhomogeneous boundary conditions, the representation (a*) remains valid. Moreover, it can be easily shown that the results (iHiv) hold in that case as well. See Ladyzhenskaya (1969). D Bibliographic notes Section 2.1: The calculus of mappings between infinite-dimensional linear spaces has an interesting history. Early references include Volterra (1930), Hadamard (l903), and Frechet (1906). More recently the following works have proven interesting: Dieudonne (l960), Nevanlinna (1957), Hille and Phillips (1957), Michal (1958), and Cartan (1970, 1971). Early discussions of the derivative are found in Gateaux (1906) and Frechet (1925). The book by Ljusternik and Sobolev (1961) contains a more-up-to-date treatment. In his thesis Goldring (1977) has completed the first steps of a Hodge decomposition theorem for nonlinear operators when regarded as differential one forms. Section 2.2: The books of Krasnoselski (1964) and Vainberg (1964) contain careful discussions of the composition operators of 2.2A. The result (2.2.10) is from Littman (1967). The method of Schauder inversion is a formalization of a technique that recurs in the papers of Schauder mentioned in the bibliography. The duality method for the definition of abstract nonlinear operators has proven to be very effective in many different contexts and is well described in Brezis (1973), Browder (1976), and Lions (1969). Secrion 2.3: Our discussion of analytic operators is patterned after Hille (1948). The paper of Taylor (1937) makes interesting reading, while the recent work of Douady (1965) may prove of general value. Section 2.4: The systematic investigation of compact operators and their connection with algebraic topology is due to Schauder. (See his papers listed in the Bibliography.) Section 2.5: A useful survey of the basic results on gradient operators can be found in Rothe (1953), and Krasnoselski (1964). Much of this work is based on carrying over the concepts of the calculus of variations to a more formal context. The result (2.5.7) can be found in Berger (1967). Goldring (1977) has extended the motion of gradient mapping by proving various infinite-dimensional versions of the Frobenius integrability theorems. Section 2.6: Nonlinear Fredholm operators were introduced in Smale (1965). An interesting attempt to extend the index theorem of Atiyah and Singer to a nonlinear context can be found in Palais (1967). There seems little doubt that the concept of nonlinear Fredholm operator will prove important in the future development of our subject. Section 2.7: A thorough discussion of proper mappings can be found in Bourbaki (1949), while the result (2.7.7) is proved and applied in the paper Ambrosetti and Prodi (1972). The result (2.7.1I) is found in Berger (1974).

PART II

LOCAL ANALYSIS

The aim of Part I1 Here we discuss the local mapping properties of a non-linear operatorf restricted to a small neighborhood of a given point of its domain. We then relate these properties to more specific notions of theoretical and concrete importance. The basic problems to be discussed To fix notation, let f denote an operator defined in a neighborhood U(xo) of a point xo in a Banach space X with range in another Banach space Y . Then we attempt to determine the behavior of f(x) nearf(xo) in asprecise aform as possible by posing the following questions:

(i) Linearizafion problem If f ( x ) is differentiable (at xo), in what sense are the properties of f(x) near f ( x o ) reflected by the linear operator f’(xo)? (ii) Localsoluabilifyproblem If 11 f ( x o - yII is “small”, under what circumstances can we solve f ( x ) = y for x ‘‘near” x,? (iii) Local conjugacyproblem If g is another mapping with domain U(xo) and range V contained in Y and such that f - g is small (in some sense), in what cases do f and g differ by a local change of coordinates, ix., there exist local homeomorphisms (“changes of coordinates”) h,: U ( x o ) + U(xo) and h,: V - V such that f = h;’gh,. In particular, if f ( x ) = Lx + O(11~11~) near x = 0 where L is a linear operator, what properties of L and f ensure that these operators are conjugate near x = O? (iv) Sfabilityproblem In what sense are the mapping properties of the operatorfnear U ( x o ) unaffected by a small (but possibly arbitrary) perturbation f + cg (for t a small real number)? If a given property is destroyed by an “arbitrary” perturbation, can the property be preserved by restricting the class of allowable perturbations? (v) Problem concerning local structure of solutions If f ( x o ) =yo, can one give a complete description of the set of solutions ( x I f ( x ) = y o , x E U ( x o ) ) ?In particular, are the solutions isolated? (vi) Problem of nonlinear effects What features of the higher order part of the operator f, viz. f ( x ) - f ’ ( x o ) ( x - xo) - f ( x o ) , are significant in studying the local properties off near f(xo)? (vii) Problem of parameter dependence If the mapping f ( x ) = f(x, A) depends continuously (smoothly) on a parameter A, how do the local properties off change as A varies? In particular, describe the behavior of the solutions of f ( x , A) = 0 near the “bifurcation” set B=(x,A)Icokerf,(x,h) # (O),f(x,A)=O). (viii) Problem concerning construction of appropriate solutions If f(x) = y has a solution I09

110

PART 11 LOCAL ANALYSIS

x near xo when f ( x o ) - y is small, can one construct an explicit approximation x to xo such that 11 x - xoI( can be made arbitrarily small? (ix) Problem concerning irerution schemes If a sequence x, is defined by the rule x, = g(x,x,,-~, ,x,-J (where k is some finite integer independent of n, and g is a

,,

.. .

continuous map defined on X X X * . . (k times) X X, under what circumstances does the sequence (or some subsequence) converge to a solution X of x = g(x, x, . . . ,x)? Furthermore, under the hypothesis of question (viii) above, can an approximate solution be defined by a convergent iteration scheme?

The problems just mentioned arise naturally in connection with the detailed study of explicit nonlinear systems. Thus methods for obtaining approximate solutions to a given local problem IT are well known and (generally speaking) readily constructible. For example, I’Imay be “close” to a problem IT’ all of whose solutions are known explicitly, and one assumes (as a first approximation) that the solutions of IT are “close” to those of IT’. Indeed, the well-known techniques of linearization, successive approximation, averaging, undetermined coefficients, and singular perturbation are among the formal constructions for such approximate solutions. Yet the validity of such approximate solutions is often left open by these formal schemes, and in fact, often taken for granted despite much evidence to the contrary. As we shall see in the sequel, the study of local analysis sheds considerable light on such questions. For example, higher order approximations (of arbitrary order N), x N ( c ) = xo ~ ~ ~ o u ntoc the n , solution of an operator equation f , ( x ) = 0 may often be constructed with the property that fc(xN(c))= O ( r N + ’ ) .However, as was mentioned in Section 1.2B, it may happen that limN+mxJc) does not exist for any c # 0, since the infinite series xun,’ diverges. Thus the question of the validity of the approximation x N ( z ) to a true solution x(e) of f , ( x ) = 0 requires further investigation.

+

CHAPTER 3

LOCAL ANALYSIS OF A SINGLE MAPPING

In this chapter we focus attention on a fixed operatorfacting between two Banach spaces or (as in Section 3.4) two scales of Banach spaces and discuss the elementary approximation and iteration schemes that relate to the inverse and implicit function theorems. In the first section of the chapter we discuss those results that can be based on the elementary contraction mapping principle. The applications of these results to ordinary differential equations in a Banach space, singularities of mappings, and local properties of extremals of isoperimetric variational problems are then discussed. The classical steepest descent and majorant methods of successive approximation are described in the next two sections. Finally (in Section 3.4) we take up the recent extensions of the iteration schemes associated with the inverse function theorem due to Nash, Moser, Kolomogorov, and Amold.

3.1 Successive Approximations

The simplest systematic approach to answer the questions of local analysis just raised is based on the method of successive approximations for the solvability of the operator equation f ( x ) = 0. In fact all the results of this section are based on this theme. Given f E C ( u , Y), the fundamental idea of the method is to define (explicitly) a Cauchy sequence of elements x, E such that f ( x n ) + O . Then by the completeness of 5 (assumed to be the closure of an open subset of a Banach space X), x, converges to some X E a, and, by the continuity of f, f(E) = 0. The simplest case of such a construction is provided by the following.

a

3.1A

The contraction mapping principle

Given a continuous mapping A of a set S into itself, one attempts to locate the fixed points of A by defining a sequence (xo, Axo, A2xo, . . . , A%,, . . . ) for xo E S and by seeking conditions on S and A that ensure the convergence of this sequence. A simple answer is the 111

112

3


(3.1.1) Contraction Mapping Theorem Denote by S ( K , p ) the sphere of radius p and center K of a Banach space X . Suppose A maps S(K,p ) into itself and satisfies the condition that for any x , y E S(F,p), (3.1.2)

[ ( A x - Avll

< K l l x -YII,

where K is an absolute constant less than 1. Then A has one and only one fixed point x , in S(X, p), and x , is the limit of the sequence x, = A "x0 ( n = 0, 1, 2, . . . ) for any choice of xo in S(X,p). Proof: First we show that x, = A ' x 0 is a Cauchy sequence for any x o E S(X,p). Indeed, for any integers n and p , comparison with the geometric series K" K " + ' . yields

+- -

+

Hence as n + m , ( I X , + ~ - xnII+O independently of p , so that { x , } is indeed a Cauchy sequence in S(7,p). Since S(K, p ) is complete, x , + x , (say) with x , E S(X,p). Hence by the continuity of A (3.1.3)

A x , = lim A x , = lim x,+, = x,, n+m

n+

00

i.e., x , is a fixed point; and it is unique since if y , were another fixed point, then (3.1.3) would imply llx, -y,ll=

IIAx,

- AVmll

KllXm

-Yc.ll?

which is possible only if x, = y m . Of the many interesting extensions of (3.1.1) the following one is quite useful when the map A depends on a parameter p. (3.1.4) Corollary Suppose A ( x , p ) is a continuous mapping of S(K, p ) x B + S(X, p ) for some metric space B , and furthermore that A satisfies (3.1.2) for each p E B . Then the mapping g : B + xp (the unique fixed point of x = A ( x , p)) is a continuous mapping of B into X . Proof: Let p, + p, in B . Then g( &) = xpn = A(xp,, for p = p,. Hence

II g( P,) - g( P,)I

=

IIA (Xp.3 IIA b p . 9 +

p,), and similarly

P,) - A ( X p J P,)I P,) - A (xp,, PJll

IIA (xp-9

Pn)

- A < x j m ,P,)II

Kllxp" - xp,ll + ll+p,,

P,)

- A(Xp,.

Pm)II

3.1

113

SUCCESSIVE APPROXIMATIONS

so that 1

Ilg(Pn) - g ( P J I-K llA(xp,, P n ) - A (xp,, P A . Since A is continuous in P, the right-hand side above tends to 0; and the result follows. 3.1 B The inverse and implicit function theorems

We now prove the well-known Banach space analogues of the inverse mapping and implicit function theorems. Both these results are obtained together with constructive iteration schemes. The inverse function theorem gives a first answer to the basic linearizatlon question raised at the beginning of Part 11, while the implicit function theorem answers analogous problems involving parameter dependence. (3.1.5) inverse Function Theorem Suppose f is a C I mapping defined in a neighborhood of some point xo of a Banach space X , with range in a Banach space Y . Then if f’(xo) is a linear homeomorphism of X onto Y , f is a local homeomorphism of a neighborhood U ( x o ) of xo to a neighborhood of f ( x o ) . Furthermore, if IIy - f(xo)II is sufficiently small, the sequence (3.1.6) % + I = x n + [f’(xo)l-”Y -f(xn)l converges to the unique solution of f ( x ) = y in V ( x o ) . Proof: Set f ( x o ) = y o . We first attempt to determine p so that f ( x o = y provided IIy - yell is sufficiently small, or equivalently

+ p)

(3.1.7) f ( x 0 + P) = Y - Yo. Since f is C ’ at xo and f’(xo) is invertible, (3.1.7) implies that f’(xo)p R(xo, P) = y - yo. i.e.,

+

P = [ f ’ ( X o ) l - ” ( Y -Yo) - R(x0, P)l? where the remainder R(xo9 P) = f ( x 0 + P) - f(.o) - f’(X0)P = 4llPII). We show that (3.1.7) has one and only one solution for llpll sufficiently small, by proving that the operator A p = [ f ’ ( x o ) ] - ’ { y- y o - R ( x o , p ) } is a contraction mapping of a sphere S(0, c) in X into itself, for some c sufficiently small. Indeed for p and p I E S(O,c), f’(xo){AP - A P I ) = R(x0, P 1 ) - R(x0, P) = f ( x 0 + PI) - f ( x 0 + P) - f ’ ( X O ) ( P l =

s,

1

{f’(xo + tP1 + (1 - 4 P )

-f’(Xo)I(P1

-

P) dt.

- P)

114

3 LOCAL ANALYSIS OF A SINGLE MAPPING

Hence

s,

1

(3.1.8)

IIAP - APlll =

l l ~ f ’ ~ x 0 ~ 1 -I ’I lf l’ b o + tp1 + ( 1

- t)P)

-f’(xo)ll IIPI - P W . Since f is a C ’ mapping, the middle term of the last intergrand can be made arbitrarily small by choosing I1pll, ((pl(lsufficiently small; and hence for some constant K < 1 (and independent of y - yo) and sufficiently small E > 0, I1Ap - Aplll < Kllp - pill for all p, p1 in S(0, E). Furthermore, A maps S(0, E) into itself. Indeed, IIApl( = [(Ap- A(0)ll + IIA(0)ll Kllpll + provided IIy - y o ll llA(0)ll and llA(0)Il = Il[f’(xo)l-l(y -yo)ll < (1 < (1 - K ) ~ l l [ f ’ ( x ~ ) ] - ~Hence ( ( - ~ .A is a contraction nlap of S(0, e) into itself, under the last proviso. By the contraction mapping theorem (3.1.l), A has a unique fixed point in S(0, S ) , where 6 < E is chosen so small that f(S(0, 8 ) ) c S(yo, (1 - K)cJJ[f’(x0)]-’II-’). Reversing the steps in the argument, one finds that f ( x o p) = y has one and only one solution when JIy- yell and IJpIJare sufficiently small. That y depends continuously on p and hence on x = xo + p follows immediately from Corollary 3.1.4 and the obvious fact that under the operator A p = [f’(xo)]-’(y- y o - R(xo, p ) } depends continuously on y. Thus f-’(y) = x is a well-defined and continuous mapping from a sphere S(yo, 71) in Y to X.Finally, for IIf(xo) - yII sufficiently small, f ( x ) = y has a unique solution x = xo + p, where p is the limit of the sequence po = 0, p, = Ap,- I . Then

+

x, = xo

+ p,, = xo + Ap,= xo

I

+ [ f ( x o ) l - ’ [ Y -f(xo)

= x,+

[f(xo)l-l[y

= %-I

+[f’(xo)l-b

-

R(x0, P n - d l

+fl(Xo)Pn-l

-f(xo+Pn-I)l

-f(x,-1)1.

Hence x = lim,,+mx,, where x,, is defined by the iteration scheme x, = x,- 1 + [f’(xo)l-“Y - f(%-J1. Corollary Under the hypothesis of Theorem (3.1.5), f - ’ is differentiable, and ( f - ‘(yo)>’= (f’(xo))(3.1.9)

’.

Proof: If f ( x o ) = yo and f ( x o

+ x ) = y o + h, then

f - I ( Y o + h ) -f-I(yo) -f’(xo)-Ih = f’(xo)-l{f’(xo)x =

- f’(X0)-

{ f(x0 + x ) - f(x0) - f ’ ( X o ) X

-h)

1

= 4llxll) = 4llhll).

Thusf-’ is differentiable and (f-’(yo))’ = ( f ’ ( ~ ~ ) at ) -yo. ~ Next we find hypotheses so that the equationf(x, y ) = 0 can be uniquely

3.1

115


solved, locally, in the formy = g(x), where the function g is as smooth as f is. (3.1.10) Implicit Function Theorem Let X , Y , and Z be Banach spaces. Suppose f ( x , y ) is a continuous mapping of a neighborhood U of (xo,yo) in X X Y into Z, f ( x o ,yo) = 0, and fy(xo, yo) exists is continuous in x and is a linear homeomorphism of Y onto Z. Then there is a unique continuous mapping g defined in a neighborhood U , of xo, g : U l - + Y , such that g(xo) = yo and f ( x , g ( x ) ) = 0 for x E U,. Proof: For fixed x near xo, we write

f(x,y) = f , ( x o d o ) ( Y -Yo) + R ( x , y ) where R ( x , y ) - R ( x , y') = o(lly - y'll) for ( x , y ) and

( x , y') near

(xo,yo).

To solve f ( x , y ) = 0 near (xo,yo), we consider the map A 2 = y - [f,(x09YO)l-If(x.Y)=Yo - f , - ' ( X o , y o ) m , y ) .

The proof of Theorem (3.1.5) ensures that for fixed x (near xo) A, is a contraction map of a small sphere centered at yo into itself. The unique fixed point y ( x ) of A,(y), which exists by (3.1.1), depends continuously on x , by (3.1.4). Furthermore, y ( x o ) = yo and f ( x , y ( x ) ) = 0. Furthermore, y ( x ) is the only continuous function with these properties, for any other such function would necessarily be a fixed point of A d . Thus we need only set g(x) = y ( x ) to obtain the desired result. (3.1.11) Corollary If, in addition to the hypothesis of the implicit function theorem (3.1.7), f , ( x , y ) exists and is continuous for ( x , y ) near (xo,yo), then the function g ( x ) is continuously differentiable for x E U, and (3.1.2)

g'(x) =

-[f,G, g(x))l-%(.? d.))

Proot: We first establish the Lipschitz continuity of g(x). Under the hypothesis of the corollary, f(x,y) is a C' mapping near (xo,yo) and in addition for llhll sufficiently smooth and (x,y) near (xo,yo), f(x, g(x)) =f(x + h, g(x + h)) = 0. Hence expanding f(x + h, g(x + h)) about (x, g(x)) we find Ilf(x?g(x)Y

+r,k

(3.1.13)

Il[fv(x.g(x))l-%(x,

g(x))[g(x + h ) - g(x)lll = o(llhll + II g(x + h ) - g(x)ll). Since&(x, g(x)) is invertible and continuous in x:

-

g(x))h + [g(x + h ) - g(x)lll

o(llhll + II g(x + h) - g(x)ll).

Thus there is a constant M independent of h such that 11 g(x + h) - g(x)ll

6

Mllh(1. Now

(3.1.13) implies that g(x) is differentiable and that (3.1.12) holds.

Remark: If f ( x , y ) E C" near (xo,yo), then the function g ( x ) is also C". This follows immediately from (3.1.12) for n = 2, and for general n by induction based on the same formula.

116

3


3.1C

Newton’s method

We now turn to a refinement of the iteration scheme (3.1.5), namely the so-called Newton method, which allows a substantial improvement in the rate of convergence of (3.1.5). This method may be described as follows: Given an initial approximation xo to the solution of f(x) = 0, we try to find a better approximation xI = xo p I , where pl is determined so that f(xI) = 0 is satisfied apart from higher order terms. Thus assuming [f(xo)]-’ exists, f ( X d = f(x0 + PI) = f(X0) + f’(X0)PI + O ( l l P l l l ) so that PI = -[f’(xo)]-’f(xo). Continuing in this manner, at the ( n 1)th stage an approximate solution x,,+ = x,, pn+ is found by setting p,,+ = -[f’(x,,)]-~(x,,); so that (provided [f’(x,,)]-’ always exists) we find a formal solution x, of f(x) in the form x, = xo ~ ~ = I p ,A, .virtue of the Newton method is the rapid convergence of x,, + .x, That is, instead of an estimate of the form \IX,,+~ - x,II < Kllx,, - X,,-~II which yields (Ix, - xNll = O(KN), one finds exponential convergence: IJX,,+~ - X J < Kllx,, for certain absolute constants c 0 , K so that IIx, - xNII = 0 [ ( E ~ K ) ~with ” ] , eo = Ilf’-’<x,)II Ilf(xo)ll. Indeed, suppose the sequence (x,,} stays in some sphere S in X , f i s C 1and Ilf’(x) -f(y)ll < M l l x -yJI for x, y E S. Then by Taylor’s formula,

+

+

+

+

(3.1‘14)

I I j ( X n ) -f(Xn-1>

By definition, f(xk)

- f ( Xn - l) ( Xn

= -fl(~~)(x~ +x k~)

- xn-1111< MIIxn - Xn-III’.

for any k , so that from (3.1.14)

Thus (3.1.15)

IIxn+l-

XnII

< MIIY-’(Xn)IIIIXn

- xn-lI121

and we can choose K = M sup,, Il[f(x,,)]-IIl. (3.1.16) Theorem Letfbe a C ’ mapping defined on a sphere S(E, 8) of a Banach space X with range in a Banach space Y , and let xo be a n arbitrary point of S(X,8). Suppose f is such that for arbitrary x , y E S(X,6): (1)

IIf(X)

-fl(Y)ll < M,llx -A;

(ii) f’(xo) is a linear homeomorphism of X + Y . Then provided Ilf(xo)II is sufficiently small, the sequence x,,+~= x,, - [f’<x,)]-’f(x,,) is defined and converges to the unique solution x, of f(x) = 0. Furthermore, llxN x,I1 = 0 ( c 2 ” ) as N -+ 00 for some positive number e < 1, with e defined by (3.1.18’) below.

3.1

117


Proof: Proceeding as in the proof of the inverse function theorem, we set f(xo) = y o and seek an element p such that f ( x o + p) - f ( x o ) = -yo. Assuming (xo+ p ) € S(X,6 ) and expanding f ( x o ) about xo + p in the form f ( x o ) = f ( x o + p) - f ’ ( x o+ p)p + R ( x o + p, p), we find the following equation for p : (3.1.17) -yo =f’(xo + p)p + R ( x 0 + P ) where IIR(x0 + P? P)II = O(llPlI). ~3

Since f ’ ( x o ) is a linear homeomorphism, so is f’(xo + p) for llpll sufficiently small, and the operator

+ P)l-’IYo + R(x0 + P, P ) ) is thus well defined for llpll sufficiently small. We shall show: (i) B defines a contraction mapping of S(x,,S’) into itself for 8’ sufficiently small, so that the equation (3.1.17) is uniquely solvable; and (ii) the convergent iteration scheme so generated coincides with the one defined in the statement of (3.1.16). To prove that B is a contraction mapping, we first derive some estimates for the dependence of [ j ’ ( x o + p ) ] - l on p using the notation [ f ’ ( x o+ p ) ] - l = g(p). By virtue of (ii) and the identity L;‘ - L;‘ = L;’(L, - L , ) L ; ’ , BP =

-[f’(Xo

II g(p) - g(p’)II M I l l P - P’ll II g(p)II II g(P’)ll* Thus setting )I g(0)ll = C , we obtain I1 g(p)II = II g ( d - g(0)II + c CMlllPll II g(p)II

< C(1 - CMIIIPII)-l. < 6’ < min{(2M,C)-’, a},

Hence for llpll Furthermore, for p, p’ E S(xo, a’), IIBP - BP’ll

+

c

g(p) exists and Ilg(p)ll

II g(dII IIR(x, + P? P ) - R(x0 + P’)ll + II g ( p ) - g(p” llY0 + R(x0 + P‘, P”. $3

Now R(x0 + PI - R(x0 + P’) = f(x0 + P’) - f(x0 + P )

+ f’Go + P>P - f’(X0 =

!’{0f (.O -

f’(X0

+ P’)P’

+ P + 4 P ’ - P))

+ P))(P’

+ [f’(Xo + P’)

- P)

ds

- f’(X0 + P ) l P .

< 2C.

118

3


Combining the above results and using the hypotheses of the theorem, we find IIBP - BP’ll G 2CMI { llp‘ - Pl12 + IlPll llP’ - PI1 1

< 4c2M:(llroll + fM,IIPl12)IIP’ - PI1 G

KIIP’ - Pll,

where K < 1 is a constant independent of p and p’ provided 6‘ and llyoll are sufficiently small. In addition IIBpll < IIBp - B(O)II + IIB(O)II < KIIpII + CIJy,JI,which implies that llBpll G 6’ provided Cllyoll < (1 K)6’. So by choosing llyoll and consequently 6’ sufficiently small, B is a contraction mapping of S(xo, 6‘) into itself. Thus B has a unique fixed point E, E S(xo, 6 ) defined as the limit of the sequence po = 0, (3.1.181

&+I

=

4%=

+ P,)I-’{~o + R ( x 0 + Pn, P,))

- [f’(Xo

= - [f’(Xo

7 f(x0 + P,)

+ PJl-

- f’(X0 + P J P ,

1

+ P,). Setting x, = xo + p,, (3.1.18) becomes the classic Newton iteration scheme x,+, = x, - ( f ’ ( x , ) ) - ’ f ( x , ) mentioned in the theorem. Setting x , = limn+, x,, the estimate fgr,IIx, - xN(I follows since by the inequality (3.1.15), I J X ~ +-~ xNll = O ( K Z ). Indeed, =

IIXN+I

Pn

- f’(X0 + P,)f(.o

- XNIl G K I b N - xN-1112

*

*

*

< g 1 + 2 + .“

+2N-’

llx,

- XOllZN

= O(K2”(1XI - XOIIZN).

Then if one sets (3.1.18’1

E = Kllx1 - XOII = ~ll[f’(xo)l-lll Ilf(x0)lL we find by (3.1.15) and the fact that m

IIxw -

)lll ds

{ K + Illf(s,

xo)Ill)a;

while for x , y E X(xo, R ) , ( ( ( A x- Aylll < K a ( ( ( x- y ( ( ( .Thus A will be a contraction mapping of X(ao, R ) into itself provided Ka < 1 and Ka + Illf(s, xo)lIla < R. Clearly these inequalities hold simultaneously if 0 < a < 1/K and a is chosen sufficiently small. (ii): Clearly the mapping A defined in (i) depends continuously on xo, so that by (3.1.4), the fixed point of A , x ( t , xo), depends continuously on xo in the topology of C ([0, a],X).To prove the more precise result, we note that if x ( t , xo) and x ( t , y o ) denote the solutions of (3.1.21with respect to the initial conditions xo and yo, then Ilx(t1 xo)

- x(t,yo)ll 6 11x0 -yo11 + &'IlAs, x(s,

20))

-f(& x(s7yo))ll ds

G llxo-Yoll + K/u'llx(s. xo) - x(s,.Yo)ll

ds.

Then denoting the left-hand side of the above inequality by w ( f ) , we find that w ( t ) > 0 and w ( f ) satisfies the inequality (3.1.26)

3.1

121


Consequently, for any interval [0, TI on which x ( t ) exists, w ( t ) G llxo- yolleKr. Indeed, multiplying by e - K r , we find dt ( e C K r / o r w ( s d) s )

(

< e-K' w ( t ) - K/o'w(s)

ds)

< llxo-yolle-".

Integrating this last inequality from 0 to T , we obtain

So from (3.1.26), we obtain (3.1.24).

Next we prove some results on the continuation properties of the solution x ( t , xo) of (3.1.21). (3.1.27) Theorem Suppose f(t, x) is a continuous function defined on R ' X X which is locally Lipschitz in x for fixed t . Then the solution x ( t , x,,) of (3.1.21) can be uniquely extended as a solution of (3.1.21) to a maximal interval [0, A ) . If x ( t , xo) exists on the interval [0, p), while limtrp x ( t , xo) exists and is finite, A > p. Prool: First, we show that any solution x ( t , xo) of (3.1.21) defined on the interval [0, y ) is unique. Suppose x ( f , xo) and y ( t , xo) are two solutions of (3.1.21) defined on [0, y), and let J = ( r l t E [0, y ) such that x ( t , x,,) = y ( t , x o ) ) . Certainly by (3.1.23), J is not empty. We shall show that J is open and closed in [0, y), so that since [0, y ) is connected, J = [0, y). The set J is certainly closed since both x ( t , xo) and y ( t , xo) are continuous. To show that J is open, suppose yo E J, then by the local uniqueness result (3.1.23), there is a 6 > 0 such that the system

has the unique solution x ( t , x ( y o , xo)) = x ( t + yo, xo) for It\ < 6. Consequently, the interval 6, yo - 6) E J , and J is open. Now we demonstrate the existence of the maximal interval [0, a) of existence of x ( t , xo) satisfying (3.1.21). Let & be the sets of pairs ([0, 6J, x ( t , x o ) ) such that x ( t , xo) satisfies (3.1.23) on [0, 6J. Clearly by the uniqueness result of the above paragraph, for any two such pairs ( [ O , 6J x d t , xo)} and ( [ O , 6J x 2 ( f . x o ) } , x d t , xo) = x2(t, xo) for f E [O,min(6xl,aXJ. Let Q = supXs66,. Then on [0, a),there is precisely one function x ( t , x,,) satisfying the initial value problem (3.1.21), and the interval [0, a) is the desired maximal interval. Finally, if x ( r , xo) exists on [0, /3) and lirnrrS x ( t , xo) = X exists and is of finite norm, we can apply the local existence theorem (3.1.23) to the initial value problem d x / d f = f ( x , r), x(0) = X, and assert the existence and uniqueness of its solution :(r, X) for some open interval (- 6, 6 ) about t = 0 . Again by (3.1.23), with x ( /3, xo) = E, (yo

+

:(?,

x) = x ( t , x ( P , x o ) ) = ? ( I + P, xo)

for f E (- 6, 6). Thus x ( t , xo) can be uniquely extended to the interval [0, /3 of (3.1.2 1).

+ 6) as a solution

For finite-dimensional Banach spaces, Theorem (3.1.23) can be improved by weakening the assumption of Lipschitz continuity of f(t, x) to just continuity itself. A quick proof of this result can be obtained by application

122


of the Schauder fixed point theorem (2.4.3). Consider the initial value problem (3.1.21), where we now suppose that x ( t ) is an N-vector and f(t, x) is a continuous N-vector function of t and x . Then we prove (3.1.28) Peano’s Theorem Under the hypotheses just mentioned, the initial value problem (3.1.21) has at least one solution x ( t , x,,) on the interval [ - T, T ] provided I TI is sufficiently small. Prool: Without loss of generality we may suppose that xo = 0. Then a solution of (3.1.21) can be found by solving the integral equation

(3.1.29)

x ( t ) = /,>(s,

x(s)) h,

x ( t ) ERN.

Denoting the integral on the right-hand side of equation (3.1.29) by Ax(r), and setting

we find that for any continuous N-vector x ( t ) defined on [0, TI with Ix(r)l < M over [0,TI,

Consequently, with C,[O, TI denoting the Banach space of continuous N-vector functions defined on (0, TI with sup norm, we find that A is a bounded mapping of C,[O, T ] into itself and that, as before, a solution of (1.3.21) is a fixed point of A in C,[O, TI.We obtain a fixed point of A by requiring that IT1 be sufficiently small and invoking Schauder’s fixed point theorem (2.4.3). To this end, note that by (3.1.30), A maps the sphere Z , = (XI llxll < M, x E C,[O, TI) into itself provided IT1 < M / K M .Thus in order to apply Schauder’s theorem it suffices to prove that A is a (continuous and) compact mapping on Z , c C,[O, M/K,]. The continuity of A follows immediately from the continuity of f(s, x(s)). To verify the compactness of A, it suffices by virtue of (3.1.30) and (1.3.13), to prove the equicontinuity of the vectors A ( t ) for x ( t ) E Z , . To this end, we note that

Hence the requisite equicontinuity property is verified, so that the desired fixed point is obtained and the theorem is established.

3.1F Application to isoperimetric problems

Many questions concerning gradient mappings G’(x) E M ( H , H) (H, a Hilbert space) can be phrased as follows: Find the extremals x of the antiderivative G ( x ) of G’(x) over a constraint set C. We refer to such problems as abstract isoperimetric problems. Here we shall investigate the operator equations satisfied by such extremals by using the results established so far in this chapter. As an application of Peano’s theorem (3.1.28), we now establish the following result for abstract isoperimetric variational problems. (3.1.31) Let H be a Hilbert space and suppose uo is an extremum of the

C ’ functional Go(u) subject to the constraint C = { u I G i ( u ) = ci

123

3.1 SUCCESSIVE APPROXIMATIONS

.

( i = 1, . . , N), where the numbers ci are constants}. Then there are

numbers Xi (not all zero) such that N

(3.1.32)

AiG,.'(uo) = 0, i=O

where G,'(x) denotes the Frechet derivative of G i ( x ) at xo. Prool: We argue by contradiction, by assuming that the vectors G;(u0) ( i = 0,. . . , N) are linearly independent. Let the extreme value of Go(u) on C be co, then we show that if (3.1.32) is never satisfied, we can find a curve u(f) E C for It1 sufficiently small with u(0) = uo such that Go(u(t)) = co + f. Since r can be positive or negative, this contradicts the facts that uo is an extremum for Go on C. To this end, let u(r) = uo Zy-ouj(r)y, where the real-valued functions uj(r) and the vectors wj are to be determined such that

+

(3.1.33)

u,(O) = 0,

G,(u(r)) = co

+ I,

G i ( u ( f ) )= ci (i = 1, . . . , N).

Assuming the w, are given, we can find the functions u,(r) satisfying (3.1.33) provided that we can solve the initial value problem

(

ci

(3.1.34)

N

u0

+ ,2 u,(r)wj

( i = 0,. . . , N ;

= yi

= 01,

J'o

where yo = 1 and yi = 0 for i > 0. Simplifying (3.1.34) we can rewrite this initial solution to the problem in the vector form du

@ ( u ( r ) ); i;= y,

(3.1.35)

4 0 ) = 0,

.

+

+

where u(r) = (uo(t), . . . , uN(r)),y = (1, 0,. . , 0), and @ ( u ( r )= (ag)is the (N 1) X (N I) matrix with entries av = (GJu, + u ( f ) .w), y) with w = (wm w,,. . . , wN). Now by Peano's theorem (3.1.28), (3.1.35) and consequently (3.1.33) have solutions provided the matrix @ ( u ( f ) )has an inverse, for 111 sufficiently small, that depends continuously on u(f). Clearly this will be the case provided det I@(u(0))1# 0. Thus we shall choose vectors y such that this determinant is different from zero, making use of the fact that the vectors G;(u,,) (i 0,. . . , N ) are linearly independent, by assumption. In fact with wj = Gi(uO), det l@(u(O))l = det l(G;(uo), Gi(uo))l# 0. If not, the system of linear equations

-

2 p , ( ~ ; ( u ~ ci(u0)) ), =o

(i = 1, . . . , N )

i

4

would have a nontrivial solution (say). Then multiplying the above equation by summing we find that for pi = fi (say), N

11 ,x&G/(uo)II= 0 1-0

pi and

N

which implies

x

i-0

&G/(uo) = 0.

.

Since the vectors G,'(u,,) are linearly independent, we find that = 0 ( i = 0,. . , N).Thus since det 1@(u(O))1 # 0, the curve u(t) E C exists for If1 sufficiently small, and therefore Gdu(r)) co f. This fact is the desired contradiction.

-

+

Remark: The result (3.1.31) holds for Banach spaces X since the (9) elements can be such that det I &?(a(0))l # 0. A related but somewhat more general result (for a possible infinite number of constraint equations) can be stated as follows:

124

3


(3.1.36) Theorem Suppose G is a C ’ mapping of a Hilbert space H into a Hilbert space H , such that for some xo, G’(xo) maps H onto H , . Then if xo is an extremal of a C functional F ( x ) restricted to the set M = {XI G(x) = 0}, there is an element h , E H I such that xo is a critical point of the unrestricted functional F ( x ) - (G(x), h’). Proof: Let T = { x I G ’ ( x o ) x = 0 ) . We first show that for arbitrary x E T , we can write an elementy E M in the formy = xo x + g, where g E [TI* and llgll = o((lxll) as llxll + O . To this end, we apply the implicit function theorem to the operator equation

+

e ( x , g)

= G(xo + x + g) = 0.

Now (regarded as a linear operator) the partial derivative 8,(0, 0) = G ‘ ( x o ) maps T I injectively onto HI,and so by Banach’s theorem (1.3.20), G’(x,) (restricted to T) is invertible. Thus the implicit function theorem implies that the equation 8 ( x , g) = 0 has a unique C ’ solution g = g(x) E T I for ( ( ~ ( sufficiently 1 small. To prove that (1 g(x)l( = o ( l ( x l ( )as ((x(( +O, we note for small r and fixed x E T that G ( x o t x + H ( t ) ) = 0,

+

where i ( t ) E T* is a C ’ function of t . Differentiating this equation with respect to t and setting r = 0, (3.1.37)

G‘(xo)x

+ G‘(xo)$(0)

= 0.

Since x E T, G‘(xo)x = 0, and (3.1.37) then implies that $(O) = 0 (since G ’ ( x o ) restricted to T * is invertible). Consequently, by (3.1.12), IIg(x)ll = 11 g(l)ll = o(llxll). Now we observe that for h E [TI* the expression f ( h ) = ( F ’ ( x o ) , h) is a well-defined bounded linear functional on T I , and therefore on HI since C ’ ( x o ) is a linear homeomorphism of T* onto HI. Consequently, there is a fixed element h , E H I such that ( F ’ ( x o ) , h) = (y, h,) for each y E H,. Thus, for each h E T* withy = G‘(xo)h, (3.1.38)

( F ’ ( x ~ ) h, ) = ( G ’ ( x o ) h , h i ) .

Finally, for arbitrary h E M ,h = n + m, where n E T I and m E T. Clearly G’(x,,)m = 0. On the other hand, by virtue of the results established in the first paragraph, h ( t ) = xo tm g(r), where I(g(r)ll = o(ltl), so that for arbitrary m E T d ;i; F(h(t))l,-o = ( F ’ ( x o ) ,m ) = 0.

+ +

Hence (3.1.38) holds not only for h E T I but for all h of the unrestricted functional F(x) - ( G ( x ) , h,).

E H.Thus xo

is a critical point on H

As a final example, consider the critical points of the C2 functional F ( x ) restricted to the hypersurface 9TL = {XI G(x) = const.} of a Hilbert space H.If G’(x) f 0 on %, then a critical point xo of F satisfies the equation

(3.1.39)

F’(xo) - AG’(xo)= 0,

where A =

( F ’ b O ) ? G’(x0))

llG’(xo)l12 This second variation denoted S2F(xo, v ) is a quadratic form defined on tangent vectors to the hypersurface %, by means of the formula S2F(X,,

0)=

d2 dt2

- F(v(t))l,=o.

3.1 SUCCESSIVE APPROXIMATIONS

Here

v(t)

125

is a C ’ curve on % passing through xo such that d

u(t)l,,o = o

with

(0,G’(xo)) =

0.

We now compute this second variation of F at xo relative to %. In fact, we have the following simple formula (3.1.40) The second variation of F restricted to % on a Hilbert space H can be written (3.1.41) S’F(x0, 0) = ([F”(xO)- h G ” ( x o ) ] ~o),,

where

(0,C’(xo))=

0 and A is given by (3.1.39).

+ +

Prool: By arguing as in the proof of (3.1.31), the arc x ( t ) = x to a ( t ) G ’( x ) lies on % if x does, where a f t ) is defined as the solution of the initial value problem (3.1.42)

~ ( 0=) 0.

~ ’ ( 2 )= - ( G ’ ( x ( t ) ) , u ) / ( G ’ ( x ( t ) ) , G’(x)).

Moreover in deriving (3.1.41) it suffices to consider arcs of the form x ( t ) . Now ( d / d z ) F ( x ( t ) ) = ( F ‘ ( x ( t ) ,x ’ ( t ) ) and (3.1.43)

d2 dt

-F ( x ( r ) ) =

By choosing (u,

( F ” ( x ( t ) ) x ’ ( r ) x, ’ ( t ) )

+ ( F ’ ( x ( r ) ) ,x ” ( t ) ) .

G ( x ) ) = 0,a‘(0) = 0 so that x’(0) = u. Thus

d2

- F ( ~ ( t ) ) l , - o= ( F ” ( x ) u , U) + ( F ’ ( x ) , a ” ( o ) G ’ ( ~ ) ) . dr Now at a critical point xo, F’(xo) = hG’(x,), implying that (3.1.44)

A2F(x0, U) = ( F ” ( x ~ ) u U), + A ~ ” ( O ) l l G ’ ( x o ) l l ~ ~

On the other hand, to compute a”(O), we note that (G‘(x(r)) - G’(x), u) = t (G”(x)u, u) + o(r). Thus from (3.1.42), since a”(0) = lim(u‘(r)/t) as t -0, ~“(0) = (G”(x)u, u)/llG‘(~)11~. Finally, from (3.1.44), we obtain S2F(xo, u) = ( ( F ” ( x , )

-

hG“(x,))u, u)

with (u, G’(xo)) = 0.

and h given by (3.1.39).

3.1G Application to singularities of mappings

The notion of singular points and singular values of a C ‘ mapping f between Banach spaces was introduced in Section 2.6. This notion represents a direct generalization of the finite-dimensional idea described in Section 1.6. Thus, it is natural to extend the main results summarized in (1.6.1) to an infinite-dimensional context. We begin by proving a useful analogue of Sard’s theorem due to Smale (1965). (3.1.45) Let f be a Cq Fredholm mapping of a separable Banach space X into a separable Banach space Y:Then, if q > max(indexf, 0), the critical values of f are nowhere dense in Y. Prool: Since X has a countable basis and nowhere dense sets are closed under countable

126


unions, it suffices to prove the theorem locally. To this end, we first prove that a Fredholm mapping is local!y closed, i.e., there is a neighborhood N(xo) of xo for any xo E X such thatfl, is closed. Indeed, since f’(xo) is a linear Fredholm map, we can write X as the direct sum X Kerf’(xd CB XI; and an arbitrary element of X, x = (1, u) with z E Kerf’(x,) u E XI. Now the partial derivative fi(z, u) maps XI onto a closed subspace of Y for all x = (z, u) near xw Thus by the implicit function theorem, we can find an open neighborhood D , CB D , of xo in Ker F’(x,,) CB XI such that 0, is compact and f restricted to z CB D, is a differentiable homeomorphism onto its image. Now let f ( x i ) = yi + y for xi = (zi, ui) E D , Q D,. To show that f is locally closed, we show that xi has a convergent subsequence. Since D , is compact, we may assume that zi+Z; and since f(Z, ui)+y, even that zi = Z. However, as already mentioned, f restricted to Z X D, is a homeomorphism. Consequently, ui + 5, so that ( x i ) has a convergent subsequence. By (2.6.8), the critical points off are closed; and sincefis locally closed, it suffices to prove that for any xo E X and any neighborhood O [ f ( x o ) ]off(xo) in Y, there is a regular value off in 0 [f(xO)]. Indeed, in this case the critical values off would be nowhere dense in Y. To this end, we use the finite-dimensional result (1.6.1(i)). Since dim coker f ’ ( x o ) < rn, Y = cokerf’(xo) CB Y,, and there is a canonical projection P : Y+cokerf’(x,). Now cp(z) Pf(z,uo) is a Cq mapping of Kerf’(xo)Q {u,) +cokerf’(xo), so that Sard‘s theorem (1.6.1(i)) implies the existence of a regular value zo for cp in P ( O [ f ( x o ) ] ) .Lety E P - l ( z o ) n 0 [f(xO)],then y is the desired regular value.

-

-

As a useful consequence of this result, we prove a result implying that any problem defined by a Fredholm operator equation of negative index is not well posed.

(3.1.46) Let f : X + Y be any Fredholm map of negative index. Then f(X) contains no interior points; i.e., if f ( x ) = y o is solvable in X, then there is a y arbitrarily near yo such that f ( x ) = y is not solvable in X. Prooh Iff(X) contained interior points, by (3.1.45) there would be a y in the range off such thatf’(x) is surjective for some x Ef-’(y). Then indexf’(x) = dim Kerf’(x) > 0, contradicting the fact thatf(x) has negative index.

Addltlonal remarks on nonllnear Fredholm operators: This result demonstrates that nonlinear Fredholm operator equations of negative index are not well posed. More precisely, for operator equations, arising naturally in mathematical physics (say), solvability of an equation of the form f ( x ) = g should not depend on the precise nature of g. Indeed precise knowledge of g is often impossible, due to experimental error or some analogous cause. Another useful consequence of (3.1.45) for C4 Fredholm maps of positive index r is the fact that if q > r for almost all g E Y, the set of elements S = ( x I f ( x ) = g} is a submanifold of X of dimension r or is empty. We now extend Morse’s theorem (1.6.1(ii)) to cover the case of critical values of smooth functionals F ( x ) defined on a reflexive Banach space X. This case is not covered by (3.1.45) since when regarded as a mapping

3.2 THE STEEPEST DESCENT METHOD FOR GRADIENT MAPPINGS

127

from X + R 1 , F ( x ) may not be Fredholm. Indeed, if F’(x) = 0, the set S = {hl(F’(x),h ) = 0} is sometimes infinite dimensional. We shall prove (3.1.47) Suppose F ( x ) is a real-valued functional of class C“ defined on a real, separable, reflexive Banach space X, such that F’(x) is a (nonlinear) Fredholm operator from X + X * (the conjugate space of X). Then the critical values of F ( x ) have zero Lebesgue measure (on R’) provided in > max(dim Ker F ” ( x ) , 2). Prool: Let xo be a critical point of F ( x ) . We shall show that ( 0 ) the criticalpoints in an open neighborhood Ox, of xo coincide with the criticalpoints of a C” real-ualued function defined on an open neighborhood of a point in R m , m max(Ker F”(x,,),2). Then applying the finitedimensional result (1.6.1(ii)), we find that the critical values associated with critical points of F ( x ) near xo have Lebesgue measure zero. Let C denote the set of critical points of F ( x ) in X. C can be covered by neighborhoods of the form Oxon C, on each of which F(O,, n C) has measure zero. Since X is separable, the covering U x E C ( Ox, n C ) has a countable subcovering. Thus F ( C ) has measure zero since the countable union of sets of measure zero also has measure zero. Thus it remains to prove ( 0 ) . To this end, since F’(x) is Fredholm, we can write X = Ker F”(xo)@ X,, so that x E X can be uniquely written as x = z + x,, where z E Ker F”(xo) and x 2 E X,. In the same way, we can write X* = coker F”(xo)63 A’;. Also F’(z, x,) = (P,F , P,F’) ( f l ( z , x J , f i ( z , x 3 ) where f l and f , denote the partial derivative operators relative to this decomposition and PI and P2 denote the canonical projections of X*+coker F”(xo) and X*+ Xy, respectively. Then L = P,F;(x) restricted to X, is one-toone and onto, and so invertible by (1.3.20). With this notation, we define a C ’ homeomorphism h : ( I , xJ+(E, i Jnear x,, by setting h ( z , x 2 ) = ( 2 , L - %(z, xJ). Then since the critical points of cp(E, 2,) = F ( h - ’ ( Z , 2,)) near h(xo) are solutions of cp’(i,2,) = F’(h-’(i, ?,))h’-’ = 0, the critical points near h(xo) of cp will be in one-to-one correspondence with the critical points of F near xg. The fact that h is a diffeomorphism is easily justified by the implicit function theorem. Next, we note that the critical points of cp(Z, 22 all belong to the subspace Ker F”(xo) so that the critical points of F ( x ) are in one-to-one correspondence with those of l(Z, 0). Indeed, if ( Z , XJ is a critical point of cp, and h ( z , x2) = (Z,XJ, then f l ( z , xJ = f2(z, X J = 0. Thus X, = 0, by the definition of h given above. Finally, we note that the functional cp(i,0) is of class C” if F ( x ) is. This follows from the fact that if F ( x ) is C” ( m > 2), then h is of class C ” - ’ . Indeed cp’(E, 0) = (fl(l-’(2, 0))+ f2(h-’(2, O))h‘-’. Since F2(h-’(Z, 0) = 0 by definition, cp’(Z, 0) = F l ( h - ’ ( i , O))h‘-’ belongs to C”-’, which implies that cp(i,0) is of class C ” . Thus ( 0 ) is established with {(i, 0) as the desired real-valued function, and so the theorem is proven.

3.2 The Steepest Descent Method for Gradient Mappings

For gradient mappingsf(x) = grad F ( x ) of a Hilbert space H into itself, the method of successive approximations just discussed can be supplemented with alternative techniques. Thus, for example, there are iteration schemes for the solution of f ( x ) = 0 (say) not involving the explicit

128

3


computation [ f ’ ( x ) ] - ’ at any point. Perhaps the best known of these is the method of steepest descent, due to Cauchy. This method consists in solving the initial value problem dx - = -f(x), x ( 0 ) = xo with grad F = f. (3.2.1) dt One easily shows that (along a solution x ( t ) of (3.2.1)) F ( x ( t ) ) decreases as t + 00. Provided the solution of (3.2.1) exists for all t , one attempts to show that l h + , x ( t ) = X exists and is a solution of f ( x ) = 0. The convergence of the method is in question, and we now take up this problem. 3.2A

Continuous descent for local minima

If F ( x ) possesses a strict relative minimum at some point x,, then the method of steepest descent is quite easily justified by the following (3.2.2) Theorem Suppose F ( x ) is a C 2 real-valued functional defined on a sphere S ( x o , r ) of a Hilbert space H , and suppose that for some absolute constant A > 0 (3.2.3) ( F ” ( x ) y , y )> A llYll2 for x E S(xo, r ) and y E H . Then provided ~ ~ F ‘ ( x o ) < ~ ~r ,/ Athe initial value problem (3.2.1) has a unique solution defined for all t , h+,x ( t ) = x, exists and is the unique minimum of F ( x ) in S(xo, r ) as well as the unique solution of f ( x ) = 0 in S ( x o , r). Furthermore, we have the following estimate for the rate of convergence of x ( t ) + x,, (3.2.4)

Ilx(t) - x,IJ

= O(ePAf).

Proof: First we note that the initial value problem (3.2.1) has one and only one solution for small t by virtue of (3.1.23). To ensure that x ( t ) stays in S(xo, r ) and that 11 dx/dtll -+O as t + 00 so that Ilf(x(t))ll +O, we argue as follows. Along a solution x ( t ) of (3.2.1), d (3.2.5) F ( x ( t ) ) = ( f ( x ( t ) ) ,x ’ ( t ) ) = - Ilx’(t)l12.

Hence F ( x ( t ) ) decreases as t increases. Also, d2 F ( x ( t ) ) = - 2 ( x ” ( t ) , x ‘ ( t ) ) = 2 ( F ” ( x ( t ) ) x ’ ( t ) ,x ’ ( t ) ) dt2

d dt Hence F(x(t)) = g satisfies the differential inequality g” + 2 4 ’ > 0. Consequently, by (3.2.5), ( d / d t ) F ( x ( t ) ) > - Ilf(xo)112e-2”‘ and so

> 2A(Ix’(t)ll2 = -2A

(3.2.6)

Ilx’(t)ll 6 Ilf(xo)lle-“‘.

- F(x(t)).

129


On integrating, we find I l x ( t ) - xoI/ < Ilf(x,)ll/A so that x ( t ) E S(x,, r ) for all t , so that the solution of (3.2.1) exists for all t . In the same way for 0 < t < t , , I l x ( t , ) - x(r)ll Q IIf(xo)llA -‘eCAt, so that for any sequence t, 4 00, x ( t , ) is a Cauchy sequence and consequently x = limtn-,, x(t,) exists in S(x,, r ) and x, is independent of the sequence t, chosen since clearly Ilx(t) - x,/I Q Ilf(xo>llA - ‘ e - A ‘ . (*) Also (3.2.6) implies that IIf(x(t))ll = Ilx’(t)ll +O, and hencef(x,) = 0. The uniqueness of x, follows from (*) since iff vanishes at x, y E S(x,, r), and x ( t ) is the line joining x a n d y , by (3.2.3)

The fact that F ( x , )

= min,(,o, r )

F ( x ) is unique follows similarly since for

x E S(x,, r), f ( x , ) = 0 and

F(x) - F(x,) =

s,

1

( f ( x m+ s ( x

-

x,)),

x - x,)

ds

3.28 Steepest descent for isoperimetric variational problems

It is useful to extend the result (3.2.2) to the case of abstract isoperimetric problems as described in the preceding section. To this end we consider a C 2 functional F ( x ) restricted to the hypersurface C defined by setting G ( x ) = const. Assuming G ( x ) is sufficiently smooth and the constraint set C is arcwise connected, we prove (3.2.7) Theorem Suppose F ( x ) and G(x) are C 2 real-valued functionals such that G’(x) # 0 on the constraint set C? = ( x I G(x) = const.), and the “formal” second variation of F at x relative to E? defined by (3.1.41) satisfies the following inequality for an absolute constant A > 0 and all x E C?, E e n ( x I llx - xo/I G Ilf(xo)ll/A) (3.2.8)

S 2 F ( x , u)

> Al/tl12

for (u, G’(x)) = 0,

then the solution of the initial value problem

- F ’ ( x ) + X(x)G’(x),

(3.2.9)

~ ’ ( t=)

exists for all f

> 0, lim,,,x(t)

X(X) = ( F ’ ( x ) , G ’ ( ~ ) ) / I I G ’ ( x ) ~ ~ ~ ,

x(0) = xo, = x,

exists and is the unique minimum of F ( x ) in

Prool:

-

e,.

Let x ( f ) denote the solution of (3.2.9), which certainly exists for sufficiently small f const., and moreover x ( f ) E C,. This fact implies x ( r ) exists as a solution of (3.2.9) for all 1. Indeed, if x ( f ) existed

by (3.1.27). We show that along x ( f ) , F ( x ( f ) )decreases while C ( x ( r ) )

130


-

only for the maximal time interval 148). the last statement together with (3.1.27) implies @ 00. Moreover, the following simple computations hold: [ F ( x ( t ) ) ] ’ = ( F ’ ( x ) - h(x)G’(x), ~ ‘ ( t )=)

- Ilx’(f)l12,

[ F ( x ( f ) ) l ” = ( F ” ( x ) x ’ ( f ) ,x ’ ( f ) ) + ( F ’ ( x ) , x ” ( r ) ) , (by virtue of the definition of X(x)). [G(x(r))]’ = 0 Consequently, F ( x ( f ) ) is decreasing along x(r), and [ ~ ( x ( r ) ) ] ”= S 2 ~ ( x x’(r)) , = S 2 ~ ( xx’(r)) ,

- (x’(r), x”(t)) +( i [ ~ ( x ( r ) ) l ” ) ,

so that [F(x(f))]“= 2S2F(x, x’(f)). Thus (3.2.8) implies that g(f) = F ( x ( f ) ) satisfies the differential inequality g”(f) + 2Ag’ > 0. Consequently, as in (3.2.3) IIx(f) - xoll < Ilf(xO)ll/A so that x ( r ) E C,. Now repeating the argument in (3.2.3) we find that x, = lim,-+, x ( t ) exists; F‘(x,) X(x,)G’(x,) 0, so that x, is the desired local minimum of F(x) restricted to C,. To prove x, is the desired unique minimum we prove for any xo E C,,

-

F(x0) > F(x-1 + ’)(IIxo - xmllh where q ( f ) > 0 for f > 0. To this end we let x(0, p) be any C ’ curve in C, joining xo and x,, and let x ( f , p) be the solution of the initial value problem (3.2.9) with initial value x(0, p). Then using (3.2.8) we find, setting x,, = ( a / a p ) x ( f , p),

(d/df)11x,,112= 2(x,,, x,,,) = -2(F”(x) - h(x)G”(x)x,,, x,,)

< -2A(1~,,11~. is a decreasing function of Thus IIxp(JeA’

IP(L 1) - X ( L 0111

F(x,), [F(x(t))l’= - Ilx’(r)l12, while [F(X(T))]” > 2A11x’(r)112. and thus from (3.2.10) F(xd > F(x,) (A? + f)llx’(f)l12. Thus x, is the unique minimum, as desired. So applying the analogue of (3.2.6) we find the desired inequality by noting that

+

Ilx’(0ll > Allx(r) - XmII > A (11x0-

Xmll

- 11x0-

x(Oll1,

so that l(xo- x(f)ll can be made small by choosing f sufficiently small since then llxo - x ( f ) ( l

< at for an absolute constant a > 0.

3.2C Results for general critical points If F ( x ) has a critical point X that is not a relative minimum, then the method of steepest descent generally does not converge to E no matter how small (Ixo- X(J> 0 may be. Consider, for example, the function F ( x , y ) = x 2 - y 2 + 4y4 defined on R2. The associated initial value problem d x / d t = - 2 x , dy/dt = 2(y - y3),(x(O), y(0))= (x,,, yo) always converges to only one of the three critical points of F ( x , y ) : (0,0) and (0, k 1). The


131

point (0,0) is a saddle point for F ( x , y ) , and one easily checks that no matter how close ( x o , y o ) is to (0,0), provided yo # 0, ( x ( t ) , y ( t ) ) = (xoe-2', g(y,, t ) ) always converges to either of the absolute minima (0, 1). Thus there arises the general problem: Does the method of steepest descent lead one to some critical point of F ( x ) (not necessarily near the initial guess xo), as in the example just given? First we prove the following simple result.

*

(3.2.1 1) Theorem Suppose F ( x ) is a C ' real-valued function defined on a Hilbert space H , such that F'(x) is Lipschitz continuous, with the properties : (i) F ( x ) is bounded from below on H . (ii) F - ' ( B ) is bounded for any bounded set B in R'. (iii) If x,,+ x weakly and {grad F ( x , ) ) + u strongly in H , then u = grad F ( x ) . Then the solution of the initial value problem (3.2.1) exists for all t and (weak) liml+w x ( t ) = X exists and is a critical point of F ( x ) . Prool: As in the proof of Theorem (3.2.3),f ( x ) is locally Lipschitz continuous, so that x ( r ) the solution of (3.2.1) exists for small r , and along x ( t ) , F ( x ( t ) ) is decreasing. Suppose that x ( r ) exists for t E [0, 1.) but not at t = r. < w . Then for 0 < ti, r2 < r.,

(3.2.12)

Ilx(r2) - x(t,)ll =

IllltZ

g

dsll < l l z l l P d F(x(s))ll dJ

On the other hand, since F ( x ( r ) ) is bounded from below for r

E [0, r.),

Combining (3.2.12) and (3.2.13),we find that if f + t., ( x ( t ) } is a Cauchy sequence in X.Thus liml+,. x ( r ) exists and is finite. Consequently, applying the local existence theorem (3.1.23)at t = r., we obtain that x ( f ) can be continuously extended for t > f . satisfying (3.2.l), contradicting the maximality of f.. Thus 1. = w . Next we show that a subsequence of ( x ( r ) } converges to a critical point of F(x). Since F ( x ) is bounded from below, (3.2.13)implies that Jgllgrad F(x(s))1I2L < w, so that llgrad F(x(r))(I +O as r + w . On the other hand, by (ii), the set ( x ( t ) ) is bounded since ( x ( r ) ) c F-'(inf, F ( x ) , F(x(0))). Therefore, { x ( r ) ) has a weakly convergent subsequence ( x ( r , ) ) as r,+ 00 with weak limit X (say), while grad F(x(t,))-+O strongly. Thus by (iii), grad F(X) 0, and X is the desired critical point.

-

A useful result, stronger than (3.2.1 l), can be obtained if we assume that all the critical points of F ( x ) are isolated.

(3.2.14) Suppose F ( x ) is a C 1 real-valued functional such that F ' ( x ) is Lipschitz continuous, defined on a Hilbert space H , bounded from below on H , and satisfying the conditions:

132


(i) all the critical points of F ( x ) are isolated; (ii) any sequence {x,} E H such that IF(x,)I is bounded and F’(x,) +0 has a convergent subsequence. Then the solution x ( t ) of the differential equation (3.2.1) exists for all t , lim,+m x ( t ) exists and is a critical point of F ( x ) . Proot: By repeating the argument of (3.2.1 I), and using the stronger hypothesis (ii) above, we may assume that x ( t i ) + x , strongly for some subsequence t,-+co. Thus it suffices to prove only that Iim,+, x ( t ) = x , exists. Indeed, otherwise there would be two spherical neighborhoods 0, and 0,centered at x , such that 0, c 0, and 8,- 0, contains no critical points of F ( x ) ; while for an infinite sequence of disjoint intervals [t,, t i + , ] there is a number c > 0 such that x ( r ) c 0, - 0, for t E [ t i , ti,.,] and llx(ti+ ,) - x(t,)ll > c. By hypothesis (ii), there is a positive constant d with I(grad F(x)ll > d for x E 8,- 0, since otherwise there would be a critical point x E 8,0,.Therefore, as r -+ co, lim F ( x ( r ) ) = F ( x ( 0 ) ) - imllgrad F ( X ( S ) ds )~~~

1-m

Thus by the above facts,

W

G F(x(0)) -

2

(cd) = - w ,

i* I

which contradicts the fact that F ( x ) is bounded from below.

3.2D Steepest descent for general smooth mappings

We end this section with a brief discussion of the applicability of the notion of steepest descent for general mappings. In his early research, Cauchy showed that the techniques just discussed can apply to the study of solvability for smooth mappings. To treat the infinitedimensional case, assume, for simplicity, that f is a Cq (nonlinear) Fredholm operator of nonnegative index r < q, mapping a real Hilbert space H into itself. Then we prove: (3.2.15) For generic p E H , the solutions of f ( x ) = p coincide with the critical points of the functional ~ ( x =) ~ l f ( x) ~11,.

Prook A simple computation shows that the critical points of F ( x ) coincide with the solution of the operator equation (3.2.16)

[f’(x)l*{f(x) - p ) = 0, where [ f ’ ( x ) ] * denotes the adjoint of f ’ ( x ) . Now by the infinite-dimensional version of Sard’s theorem (3.1.45), the singular values f(S) of f form a residual nowhere dense set. Thus for (generic) p B f(S) dim Kerf’*(x) = dim cokerf.(x) = 0.

3.3 ANALYTIC OPERATORS A N D THE MAJORANT METHOD

133

Thus if X is a solution of (3.2.16). for such generic p . f ( X ) = p . Thus the desired result is established. Thus assuming the point 0 @f(S),we can apply the previous results of this section to study the solutions off(x) = 0 by considering the critical points of the functional F ( x ) = IIf(x)l12 in each of the results of Sections 3.2A-C.

3.3 Analytic Operators and the Malorant Method

For complex analytic mappings, additional methods are often available, allowing a more complete discussion of the problems raised at the beginning of Part 11.

Heuristics

3.3A

To illustrate this point, we begin by considering the formal method of undetermined coefficients, and its justification by "Cauchy majorants." This procedure forms the main classical approach to the study of nonlinear problems and still retains an important place in nonlinear analysis. Suppose, for example, that one wishes to solve the (analytic) operator equation f ( x , A) = 0 defined on a Banach space x for small values of the parameter A, given that f ( x , 0) = 0 has the solution xo. Then to apply the method of undetermined coefficients, we postulate a solution of the form x ( A ) = xo +

c x,,A", W

where x,, E X .

I

?I=

Assuming that such a solution exists with a positive radius of convergence, one expands f(x(A), A) in the form W

f(.(A),

A)

=

2 f,,(x,,

*

. , x,,)A", *

fl=O

solves the resulting implicit systemf,,(x,, . . , , x,,) = 0 ( n = 1, 2, . . . ), and attempts to show that in a small sphere about xo that this system has one and only one solution (XI,. . . , E,,). Finally, one justifies the assumption that such a solution x(A) = xo + C,",lX,,Afl has a positive radius of convergence by finding a majorant series x*(A) for x(A). More precisely, (3.3.1) Definition x*(A) majorizes x(A) (and we write x(A) 0 is chosen so that

(3.4.2) ME'K~' Q K - @ . Then IJX,+~ - ~ ~ < 21 K - p1 " . ~ Verlflcatlon of 3(n

+ 1):

By virtue of (b) above,

IIX,+~ - xn\Ir= l[p,ll, < K S ; - ' < t K " ' + ' , provided 1 KS*+'. (3.4.3) KS;-' < ' 2 Verlflcatlon of l(n + 1): Again by virtue of (a) and (b) above, Ilf(~n+i)ll

< IIf(xn)

+ f ' ( ~ n ) ~ c+ Q(xn9

G

+ f'(xn)~cII

Ilf(~n)

PJI + IIQ(xn9 PJII

< MKs;;X+ ~ l l P € l l ~ - ~ l l P € l l ! < MK';;" + M ( 2 K - p " } 2 - p { E - I K ~ " } ~ . Now we attempt to choose E and p so that the right-hand side is at most K - p " + ' .Since (3.4.2) is automatically satisfied if (3.4.5) below is, it suffices to choose E and p so that KS;;-l = 1 KSn+' (3.4.4) T (3.4.5) MKS;;" + K -w'+'

(3.4.6)

M ( 2 K - ~ n ) 2 - P ( c - ' K s "< ' ) fPK - P " + l .

Since (3.4.4) implies that E - ' is of the order K s n ( S - ' )for , K sufficiently large, (3.4.5) and (3.4.6) will be satisfied if

(3.4.7)

sp

+ 1 < h(s - 1)

and

s( p

+ p ) < y(2 - p).

A short computation now shows that if s > 1, both relations of (3.4.7) will be satisfied if

(3.4.8)

(

0 0, has been constructed operators Tt: X , 4 X , with the properties:

+, 01 and II~(u)f(u)ll,+po Q C211ullx; Thus if f’(u): X , + Y is a bounded operator with IIf’(u)II Q C,, we choose as an approximate solution for the equation f’(u)p + f ( u ) = 0, pE = - T E L ( u ) f ( u )(which we interpret as a “smoothing” of the element L ( u ) f( u)). Then to ascertain the approximate solvability of the linear equation with this choice, we note that for u E X,, IIf’(u)Pt

+f(u>ll =

Ilf’(4{(I- TJ - ~ ) ~ ( u l f ( +f(u)II u)

= Ilf’(u)(I -

T&(u)f(u)ll

Q

Ilf’(u)ll l l ( I

- Tt)L(u)f(u>lla+o,

Q

Ilf’(.>ll{Ct

-p}IIL(u)f(u)ll,+,,+p

< c,c,[p

~ ~ o ~ l ufor~ ~p a= Po - u I .

On the other hand, llPella = II T*w4lf(u)ll, Q

Q

c5 +“II~(”)f(u)lla-o

ct +“Collf(u)ll.

Consequently, in accord with our notion of approximate solvability llpEll , 1 / E , so setting t = E , [ - p = E P / ~ we , find that the smoothing operator with properties (a)-(d) defines the approximate solvability of f’(u)p + f(u) = 0 of degree ( P o - a l ) a p ’ .

-

142


3

Typical smoothing operators are: A.

Truncation-Fourier series in d variables, x

=(x,,

. . . , xd). Indeed, if

X, is the space C' of 2 ~ p e r i o d i cfunctions u(x) of class C', over D { x I ( x i [< 2a (i = 1, . . . , d ) , x = (xl, . . . , x d ) } with then each v

E

u=

=

can be represented by the Fourier series vk ei(k'x), k = ( k , , . . . , kd). IklG P

We then define the truncation operator TNv=

2

uk ei(k'x),

Ikl< N

and note that the following inequalities hold for N

> 0:

(i) ) ) T N u J J < ~ ,c+ N , ~ + ~II4 +1' (ii) ll(Z - TN)ullcr< cN - s + d + l IlulIe+'* r . 9

Convolution operators acting on functiop with compact supporl. Let is a C" function defined on R" and vanishing outside 151 < 1 and equal to 1 for 161 < 4. Then the convolution operator

B.

+ ( x ) be a function whose Fourier transform)'#I

T p = C"+(t(X

--U))"(V)

4

satisfies the inequalities (a) and (b) above, with X , = C,*(s2),where s2 is a bounded set in Rd. Indeed, (a) follows since differentiation commutes with convolution. Whereas the second inequality follows for the same reason if we verify it with r = 0. In this case, since

-/#(z)u(x -

) dz = / + ( z ) (

u(x) - u(x

-

i )}

dz,

expanding u ( x ) - u ( x - z / n in a Taylor series,

3.4D Inverse function theorems for local conjugacy problems

In Sections 3.4A-C, we discussed an extended version of the Newton iteration scheme x,+ = xn - [ f ' ( x n ) ] - I f ( x n ) for a solution of f ( x ) = 0 near a given first approximation xo. Here we give a similar extension for the iteration scheme x , + ~= x, - [f'(xo)]-If(xn) associated with the inverse function theorem. This last scheme has the advantage that the inverse of

143

3.4 GENF.RALIZED INVERSE FUNCTION T H E O R E M S

f ’ ( x ) need be computed only at xo. The need for such an extension arises in the study of conjugacy problems for mappings such as those mentioned in the beginning of Part 11. Suppose, for example, that f and f+ a are C ’ mappings of a solid sphere of radius r and center xo, S(xo, r ) c X (a Banach space) into itself. Moreover, suppose llall small relative to f in S(xo, r ) . Then we ask if there is a nonsingular “change of coordinates” u defined on S(xo, r) (i.e., u is a diffeomorphism of S(xo, r ) into itself) such that for r > 0 sufficiently small

u - ’ ( f + a)u = f . (3.4.9) In a finite-dimensional Banach space, the rank theorem of advanced calculus implies that f and f + a are conjugate locally in the above sense if their derivativesf’(x) andf’(x) a ’ ( x ) have the same rank in a sufficiently small sphere S(xo, r’). The corresponding Banach space theorem is unfortunately more difficult, although see Note E at the end of this chapter. A formal construction for the homeomorphism u can be given in certain cases by the following iteration scheme: Write u as a perturbation of the identity u = I y . Then we define (3.4.10) u,,-Z, l l N + ‘ = u No { I + Y ~ + ~ } ,

+

+

where y N + is determined by solving (in some “approximate” sense) (3.4.11) u;ll { f + a } 0 u N + ,=f. More precisely, defining the operator F(f,u ) = u - y u for homeomorphisms u E C(S(x,, r ) , S(xo, r)), note that the following “semigroup” property holds: 0

(3.4.12)

F(f,u

0

U) =

F ( F ( f , u), u).

Thus the left-hand side of (3.4.1 1) can be rewritten

F ( f + a, uN ( ‘ + y N + I ) ) + F ( f N , ‘+YN+I)7 where fN = F(f+ a, u N ) . Assuming F(f,u ) is a C ’ function of its arguments, noting that F(f,I ) = f and q(f,I ) = I , and expanding F(f,u) about (f,I ) we find by (2.1.31) F ( f N ~ r+YN+I)=f+(fN-f)+

o(llfN-fll+

z)yN+l+

Fu(f

llYN+IIO.

Consequently, a “reasonable” approximation y N + for a solution of (3.4.1 1) is a solution of (3.4.13) Fu(f,I ) y + (fN -f) = 0. If F,(f,I ) is invertible, then y N + Iis uniquely determined and a formal solution u for (3.4.9) can be written (3.4.14)

u = N+m lim uo o u 1 o u2 o . =

*

.

lim ( I + y l ) 0 ( I + y 2 ) N-+m

’N 0

.

*

*

0

( I +yN).

144

3


Now the convergence of this formal construction is in question, and we shall discuss this question briefly here. A problem similar to that posed by (3.4.9) occurs in the transformation theory of ordinary differential equations. Consider the following system of N ordinary differential equations near x = 0 (3.4.15)

dx dt

- = f ( ~ +) a(.).

Suppose the solution near x = 0 of the reduced system d x / d t = f ( x ) is well known, whereas a ( x ) is a small perturbation of f ( x ) (for 1x1 small). Then it is natural to seek a diffeomorphism near x = 0 (i.e., a local coordinate transformation) x = U ( y ) leaving the origin fixed and transforming the perturbed system (3.4.15) into the known one dy/df =f(y). In this case, for example, periodic solutions of the perturbed system near x = 0 could be found from periodic solutions of the known ones near y = 0. Indeed, closed curves near x = 0 in the x coordinates correspond to closed curves in the y coordinates near y = 0. Now if f ( x ) is a linear map of [ W N + RN and a ( x ) = o(lxl), the problem discussed above coincides with the linearization problem mentioned at the beginning of Part 11. In this case, letting F(f, U ) = U ' - ' f U for y a differentiable homeomorphism of S(x,, r ) into itself, we again find F(f,y I o y 2 )= F ( F ( f ,yl),y2),so that the same formalism for the solution of (3.4.9) can be used to solve the problem posed by (3.4.15). Indeed (3.4.15) can be reduced to dy/dt =f(y) if we find y satisfying U ' - ' ( f + a ) U = f , i.e., F(f+ a, U )= f . (3.4.16) We shall now reconsider the iteration scheme (3.4.11). To this end, we rewrite (3.4.9) as (3.4.17)

G(x, a ) = ( I

+ ~ ) f -( f + u ) ( Z + X) = 0.

We seek a solution x for (3.4.17). By analogy with the implicit function theorem (3.1.10), we would expect a solution x to exist provided IIG(0, a)ll is sufficiently small and G,- '(0, 0) exists. Indeed, one can demonstrate that such an analogy is true even if [ G,(O, O)]- is mildly singular, by essentially studying the convergence of the iteration scheme (3.4.1 1). We begin by rewriting the iteration scheme (3.4.11) in terms of a solution of G(x, a) = 0 near x = 0. If uN+, = I x N + I= uN o { I + Y ~ + ~ ) = ( I + x N ) o ( I + y N +J, then

'

+

(3.4.18)

x ~ + ~ = ( I + x ~ ) ( I + Y ~ +I , ) -

+A.

= ( I +Yl)(I . * ( I +YN+I) - I, where y N + Iis a solution of (3.4.13). Thus in terms of G , (3.4.13) can be rewritten (3.4.19) G,(O, O h + I + aN+ I = 07

I45

NOTES

where a,+I =fN - f = F(f+ a , I

+ x),

-f

(sincef, = F(f+ a, u N ) )

that is, (3'4'19')

'N+I

= (('+))N)-')(f+

aN)(z+YN)

-f.

The convergence of the scheme (3.4.19-3.4.19) and its applications to physics and geometry are the subjects of recent books by Sternberg (1969) and Moser (1973b), and so the interested reader is referred to these for further information. NOTES A Exlstence of local solutlons of nonllnear elliptic systems

Suppose the following system of k equations in k unknowns is eNiptic on some domain Q C9" F(x, u, . . . , D"u) = 0, (*) where F and u denote vector-valued functions, with F a smooth function of its arguments. Moreover suppose q,(x) is a known smooth solution of (*). Then the contraction mapping theorem (3.1.1) can be used to prove the following result: In a sufficient/y smooth cneighborhood Ox0of any poini xo E 0,there is a smooth solution u of (*) such that sup1D'u D%,( G Czm-lnl+ofor (a(G m where a E (0, I ) and o and C are absolute consfants.The idea of the proof is to set up the desired solution of (*) as a fixed point of a contraction mapping of the Holder space C m . o ( O , Jinto itself. To this end we suppose without loss of generality that xo = 0 and uo(x) E 0 and introduce a small parameter z > 0 into the problem by setting x = cy and u ( z x ) = ~ ( yso) that by rewriting (*) in the form

(**I

LO = LO - ernF(ty,U, . . . , c - ~ D , Y o ) ,

where L is the linear elliptic operator with constant coefficients in which only the highest derivatives occur obtained by linearizing (a) about xo = 0 and uo(x) = 0. Then L is invertible and the equation (**) can be written in the form o = u - A ( € , u), where the right-hand side has Cm*"(Oxd) norm that is O ( c ) as c - 0 . See Nirenberg (1973). B lsometrlc lmbeddlng problem ?or Rlemannlan manlfolds

Let ( X k ,g) be a given Riemannian metric with metric tensor g = ( 4 )Then . we attempt to imbed 3nk as a submanifold in some Euclidean space R N in such a way that the imbedding is isometric, i.e., the metric induced on %" by this embedding is g. Thus, let z l , . . . , zN be Cartesian coordinates in R N and X X kbe smoothly (isometrically) imbedded in RN. Then referred to a set of local coordinates (xI. . . . , x k ) in ak, we require that the functions z = ( z , , . . . , z N ) satisfy the nonlinear differential system

146

3


Nash (1956) proved that this system could also be solved provided N (dimension of W N ) is made sufficiently large. He proved this result in stages, a key element of which was a forerunner of the implicit function theorem described in (3.4.1). Since this topic is well treated in several monographs, we shall not take up this topic here, but rather refer the reader to the monographs by Sternberg (1969) and Schwartz (1968). C The center problem

As a direct application of Section 3.4D, we consider the problem of showing that the analytic function

is conjugate to the linear function fl(z) = X r near z = 0 by means of an analytic change of variables u(z). Thus we seek a conformal mapping u ( z ) = 2:- ,b,z" defined near z = 0 such that

(*I

u(&) =f(utz)).

If A is not a root of unity, a formal solution can be found by setting b, = 1 and then determining b, recursively from (*) by equating coefficients of like powers of I. Explicitly, for n > 1,

(A"

- X)b" = g,,(bl,. . .

1

b"-,).

If [A( # 1, the convergence of this series can be proved by the majorant method described in Section 3.3. However, for Ih( = 1, the excluded roots of unity are dense; and, in fact, there is a dense set of complex numbers h at which the formal series diverges. Thus the convergence of the formal series for 1x1 = 1 is in question and a method more refined than the majorant method is required. C. L. Siege1 (1942) succeeded in studying the convergence of this formal series by imposing an infinite number of conditions on the number h of the form

(**I

Ihq

- 11-l

G coq2

for q = I, 2, . . .

.

(These conditions ensure that A is not well approximated by roots of unity.) We shall apply the method of Section 4.3D to prove Siegel's result.

Theorem Suppose Ihl = 1 and that h satisfies the inequalities (**). Thenf(z) = hz + f2(z) is conjugate to f,(z) = hz by an analytic change of variables. Although the proof of this result can be carried out in various ways, we sketch how it can be carried out on the basis of our discussion of Section 3.4D and refer the interested for the details to the article of Moser (1966) or the monograph of Sternberg (1969). We consider the mapping G ( ~ , a ) = ( I + z ) ( h l) (AI+u)(Z+Z)

with n ( z ) = f 2 ( z )defined on a scale of Banach spaces (A,,) and with range in another scale (E,,), both of which are Banach spaces of holomorphic functions defined on varying neighborhoods of the origin in @ I . A formal calculation of the Frechet derivative of G ( x , 0) with respect to x , evaluated at (0,O)shows that GJO, 0)u = o(k) - A@). Now the linear operator G(0,O) is invertible as a map between the spaces (A,,) and ( 8 , )if properly chosen. However, the "norm" of G,(O, 0) has a singularity of order 3 in the following sense: The formal solution of the equation o(Xr) - Xo(z) = g ( z ) is W

u(z) =

2 k-2

(Ak

-X)g/,

147

NOTES

where g(z) = Z g,,r". If g(z) is analytic on Z, the disk IzI imply that if we set II gl, = SUP^^^ 0 sufficiently small even though the linear mapping f’(x(O), 0) is not Fredholm. Both cases arise frequently in concrete problems. In bifurcation phenomena, nonuniqueness of solutions is of prime consideration, and i t is often found that such considerations, lead directly to relatively sophisticated topological techniques of solution. On the other hand, the singular perturbation problems we study generally require quite sharp analytic estimates on the norm of the linear mapping f’(x(c), c) for resolution. Both situations occur naturally in the study of explicit nonlinear eigenvalue problems of the form A x = ABx. Roughly speaking, letting A + w and setting A = I / < , it is natursl in singular perturbation theory to attempt to prove that there are families of solutions x(c) near each solution x ( 0 ) of Bx = 0. On the other hand, assuming that the linear operators A’(0) and B’(0) are both nonzero, bifurcation theory attempts to compare the solutions ( x , A) of A x = ABx and the solutions of the linear eigenvalue problem A’(0)x = AB’(0)x for llxll sufficiently small.

’

4.1 Blfurcation Theory-A

Cone

ti,

lpproach

Bifurcation theory is concerned with the b . af the solutions of the equation f ( x , A) = 0 as a function of the parameter A near a solution (x,, A,) that is also a singular point of the mapping f ( x , A,), (so that f x ( x o ,A,) is not invertible). Here f ( x , A) denotes a C ’ operator mapping a neighborhood of (x,, A,) in the Banach space X X Z into a Banach space Y . (The parameter space Z = {A} will usually be chosen as the real or complex numbers.) Thus at (x,, A,) the linear operator fx is not invertible, the implicit function theorems of Chapter 3 are not directly applicable, and 149

150

4 PARAMETER DEPENDENT PERTURBATION PHENOMENA

indeed the behavior of the solutions of f ( x , A) = 0 near (x,, A,) is indeterminate. As mentioned in Section 1.6 if X = Y = C N, 2 = R’, and f is a complex analytic mapping, one concludes that the mappingf(x, A,) is not one-to-one near x, and hence f ( x , A,) is not a local homeomorphism there. In this section we shall investigate this “nonuniqueness” phenomenon from a constructive point of view. In Section 4.2, we shall return to the more powerful qualitative methods mentioned above. I t is assumed for historical reasons that the equation f ( x , A) = 0 has a known family of solutions (x,(E), A,(€)) containing the point (x,, A,). One of the objects of bifurcation theory is then the assertion that f(x, A) = 0 has another family of solutions (x,(c), XI(€)) distinct from the family (x,(e), A(€)) near (x,,, A,) and such that (xl(c), X,(r))+(x,, A,) as A-+Ao. (See Fig. 4.1 .) In this sense bifurcation theory resembles the spectral theory for a linear operator in which case the known family (x,(c), A,(€)) is the null solution (0, A), whereas the secondary families represent linear subspaces of eigenvectors. IlXll

A

(x,(f). A,(c))

XO

A

----------

I

( X O ( ~ ) AO(f)) ,

I

I I I I

I I I

I

c h

FIG. 4.1 Usual behavior of solutions of f ( x , A) = 0 near a bifurcation point. The (Ilxll, A) graph of solutions is called a bifurcation diagram.

4.1A

Definitions and basic problems

To obtain a better understanding of bifurcation theory, let us examine the origins of the subject. In a paper published in 1885, H. Poincare attempted to answer the following questions: (a) Find the possible forms of equilibrium of a homogeneous mass of fluid (subject to gravity) when rotating about a fixed axis with constant angular momentum a.

4.1

(b)

B I F U R C A T I O N THEORY-A

151

CONSTRUCTIVE APPROACH

Determine the stability or instability of each form.

It was known that: (i) if w = 0, the only possible form is a sphere; (ii) if w is small, a family of ellipsoids of revolution M , (Maclaurin’s ellipsoids) exists and is stable; and (iii) at a certain critical number w, (although continuing to exist) this family becomes “unstable,” and a new family of equilibrium forms J , (ellipsoids with three unequal axes, otherwise known as Jacobi’s ellipsoids) becomes stable. Poincare found that these ellipsoids (initially deviating slightly from Mu), in turn become unstable at a higher critical number wl. Near w , nonellipsoidal, pear-shaped forms of equilibrium P , exist (again initially deviating slightly from J,). Poincare hoped by pursuing this argument to prove that the moon “split off” from the earth by tracing P, as a changes. (See Fig. 4.2.) Unfortunately, it was determined (after much dispute) that P , was unstable, and therefore Poincare’s argument for the origin of the moon was abandoned. The mathematical content of Poincare’s ideas, however, had a quite different fate. Poincare said that the ellipsoids J , bifurcate from M , at w,, and the pear-shaped figures P, bifurcate from J , at wl.The families { M , ) , ( J , ) , { P,) were termed h e a r series or branches. The pairs ( M W owo) , and (J,,, w , ) were called points of bifurcarion. Stability was determined by showing that the potential energy of a figure (F,, say) was a relative minimum. Poincare termed the transition of stability at (Ma,,wo) to (Ju0. wo) exchange of srubility. In the following sections we shall see how these terms apply to more general circumstances and illustrate their occurrence in many concrete problems. We now give precision to the above terms relative to the operator equation (4.1.1)

f ( x , A)

=0

for (x, A) E X x Z

(4.1.2) Definltlon A point (x,, A,) is called a point of bifurcation relative to the equation f ( x , A) = 0 if: (i) (x,, A,) lies on a curve of solutions (x,(z), A,(€)) through (x,, A,); and (ii) every neighborhood of (x,, A,) in X X Z has a solution of f ( x , A) = 0 distinct from the family (x,(c), A,(€)). In case (x,, A,) is a point of bifurcation for the equation f ( x , A)

FIG. 4.2 Poincare’s vision of the creation of the moon by fission.

= 0,

the

152

4

PARAMETER DEPENDENT PERTURBATION PHENOMENA

solutions of f ( x , A) = 0 often consist of distinct continuous curves through A,) and these curves are called branches of solutions. A (global) continuation of such a branch of solutions is a continuous curve ( x ( E ) ,A(€)) o! solutions for f ( x , A) = 0 such that Ilx(c)II llA(~)ll+ 00. A solution (2, A) of f ( x , A) = 0 is called stable if the spectrum of th_e linear has negative real part; (2, i) is unstable if f,(K, A) has a operator f,(R, i) spectral value with positive real part. This implies that the initial value problem (associated with f ( x , A) linearized about (2, i)) (x,,

+

has the property that all solutions decay if stable, but not if unstable. This notion is generally caJled linear stability theory since the spectrum of the linear operator fx(2,A) is used as a stability criterion. (A more accurate stability theory would, of course, utilize an initial value problem for the full non-linear operator.) We can state the basic problems of bifurcation theory in terms o f these definitions: (i)

Existence problem

Determine the points of bifurcation of f ( x , A)

= 0.

(ii) Structure problem Determine the complete structure of the solution set of f ( x , A) = 0 near each point of bifurcation. (iii) Continuation problem Determine circumstances under which branches of solutions admit global continuations. (This problem will be taken up in Part 111, due to its nonlocal nature.) (iv) Stability problem Determine stable branches of solutions for the equation f ( x , A) = 0 near a point of bifurcation. (v) Linearization problem What information concerning the problems of bifurcation theory can be determined from a knowledge of the derivativef,(x,, A,) at a point of bifurcation (x,, A,)? (vi) Problem of nonlinear effects What role does the nonlinearity of the map f ( x , A) play in regard to problems (i)-(iv) above? As a simple example, let U ( x , A) be a real analytic function defined on = 0 near a point (x, A,) at which V U ( x , A) = 0. If the Hessian determinant, I Ux,?(x0,A,)[ # 0, the implicit function theorem implies that there is a unique curve (x(e), A(€)) through (x,, A,). However, if I U , , , ( x , , A,)[ = 0, then the point (x, A,) may be a point of bifurcation, and in particular, there will always be a second curve ( x ( E ) , A(€)) through (x, A,) (perhaps complex-valued) satisfying the equation V U ( x , A) = 0. However, since complex-valued solutions are excluded from consideration, the problem of deciding whether or not (x,, A,) is a point of bifurcation requires further investigation.

R N X R', and consider the solutions of V U ( x , A)

4.1

BIFURCATION THEORY-A

153


A related problem (that can often be reduced to a bifuraction question, as posed here) is concerned with the structure of nontrivial solutions of the C ’ Fredholm operator equation g(x) = 0 relative to a given curve of solutions x ( t ) passing through a singular point x(0) = x,, of g. Here the term nontrioial referes to solutions of g(x) = 0 distinct from x ( t ) . To relate this problem to our discussion we set f(y, t) = g ( x ( t ) + y ) , then f(0,t ) = 0 and moreover b(0, 0) = g’(x(0)) is a linear Fredholm operator.

As an interesting example, consider the problem of finding periodic solutions of the system (4.1.3)

x +Ax

+ f(x, k) = 0,

If(x, y)I

= o(lxl

+ lyl),

near the singular point x = 0. Here x(t) is an N-vector function of t , A is an N x N (self-adjoint) nonsingular matrix that possesses k positive ei< hi < . . . < (say), and f(x, y) is a smooth function of x,y. genvalues One generally obtains a first approximation to the periodic solutions of (4.1.3) by a process of linearization. Namely, we seek periodic solutions of (4.1.3) that are near periodic solutions of the linear system

x +ilx

(4.1.4)

=0

Since this linear system possesses k distinct “normal mode” periodic families of solutions zI. z2, . . . , zk (cf. Section I.~(v)),one attempts to find at least k distinct periodic families for (4.1.3) deviating only slightly from zl. z2. . . . , zk near x = 0. To relate this problem to our bifurcation theory, it is important to introduce the period of a tentative periodic solution of (4. I .3) explicitly. (The period then plays the role of the parameter h in the general theory.) This can be accomplished by making the change of variables t = hs in (4.1.3) so that (4.1.3) becomes (4.1.3‘)

x,,

+ P ” 4 x + f(x, x,/h)]

= 0.

Then solutions of period 1 in s are solutions of period h in t . An important classic result of Liapunov in this connection is a criterion for any given periodic solution zj to be preserved by the higher order perturbation f . The result can be expressed as follows.

Liapunov’s criterion Suppose f(x, - y) = -f(x, y), then thejth periodic family of (4.1.3) is preserved if X,/hJ # integer for i # j (i = 1, 2, . . . , k ) . If f(x, y) is real analytic in x and y, the family xj(c) and its period can be written as

!j(~)

130

=

2I

n=

where a l . , , ( f=) z,(f), and

+(E)

=

2774

+ znm=I &en.

We shall show that this result can be obtained by the general results discussed here. Indeed, we shall interpret the “unusual” condition hi/hj # integer in terms of the multiplicitly of solutions of the linearized equation,

154

4


(4.1.4). In the classical proofs of this result, this condition was a necessary concommitent of the majorant method for analytic f(x, y). 4.1 B Reduction to a finite-dimensional problem

(4.1.5) Theorem Suppose X and Y are real Banach spaces andf(x, A) is a C map defined in a neighborhood U of a point (x,, A,) with range in Y such that f ( x o , A,) = 0 and fx’(xo, A,) is a linear Fredholm operator. Then all solutions (x, A) of f ( x , A) = 0 near (x,, A,) (with A fixed) are in one-toone correspondence with the solutions of a finite-dimensional system of N , real equations in a finite number No of real variables. Furthermore, N o = dim Ker L and N , = dim coker L . Proof: Set A - A, = 6 and x - x, = p, then setting - L = fx(xo,A,) we note that the equation f ( x , A) = 0 can be written in the form

(4.1.6)

LP = R ( p , A,, A)

with R ( P , A,, A) = f(x, A) - f(xo, A,) - f x ( x o , A,)

so that R ( p , A,& f(X,

= O(ll8ll)

+ dllpII + Il6ll). Indeed,

A) = [f(x, A) - f(xo, All + [ f ( X o , A) - f(Xo9 = fx(xo9 Ao)P + 4 l l P I I +

IlSll)

A011

+ O(ll6ll)-

+

Now we write p = p, p2, where p, E Ker L, p2 E XI and X = Ker L €33 XI, and recall that by (1.3.38), L has a left inverse Lo with range XI and kernel, coker L. Thus premultiplying by Lo, (4.1.6) becomes (4.1.7)

P2

= L,R(p,

+ p2, A,, A).

An application of the implicit function theorem to (4.1.7) now yields a unique solution p2 = g ( p , , A) of (4.1.7) provided llp,ll,llA - Aoll, and llpzll are sufficiently small. Since Y = Range L €33 coker L, and Lo is one-to-one on the range of L, it remains to satisfy (4.1.6) on coker L. Thus if P is the projection of Y on coker L , the solutions of f ( x , A) = 0 near (x,, A,) are in one-to-one correspondence with the solutions of

(4.1.8)

P R ( P ,+ g(pl, A), A,, A) = O -

By choosing appropriate bases for ker L and coker L the system (4.1.8) is equivalent to N , real equations in N o real unknowns. (4.1.9)

Corollary Under the above hypotheses, suppose f ( x o , A) = 0 for X near A, and

fx(xo, A) = I - AL, where I is the identity operator. If X = Ker(1- A&) @ XIand (x, A) is a solution of f ( x , A) = 0 near (x,, Ao) with x = p I g, p I E Ker(I - A,L), g E XI,then 11 gll

+

= O(llPIII).

4.1



Proot Sincef(x, A) = f ( x o , A) + fx(xo, A)(x - xo) + O(llx equation (4.1.6) can be written as ( I - AL)x

-

155

xollz),we may suppose that the

+ T ( x , A) = 0,

where T(0, A) = T,(O, A) = 0. Consequently, the equation (4.1.8) can be written (4.1.10)

h ( p , , g) = ( I - AL)g

- P T ( p , + g, A)

= 0.

Since g = g(pl) was determined by application of the implicit function theorem, (3.1.11) implies (4.1.11)

g,,h) =

PI, g ) ] - ' [ h p l ( ~g)]. ~9

We show that IIgp,(pl)ll= o(1) as JJp,JI+O, so that by the mean value theorem (2.1.19), 1) g ( p & = o(llpIII). Indeed hg = (I - AL)g - PT&, + g, A), and since T,(O, A) = 0, hg(O, 0) = I - AoL, which is invertible on X , . Consequently, by continuity, hg(pI, g) is invertible for p I and g sufficiently small and llhg(pl,g)-'ll < fll[h,(O, O)l-lll. Now Ilhp,(P,9

g)ll = IlPT,(P, + g, A)/l = 4

1)

as llplll + O . Here the convergence is uniform for A near as I(pI(1 +0 uniformly for A near A,.

A,. Thus by (4.1.11), (1 g,,,(p,))~= o(1)

4.1C The case of simple multiplicity

Clearly Theorem 4.1.5 implies that the totality of solutions (x, A) near (x,, A,) can be completely determined from the finite-dimensional system (4.1.8). This system is referred to as the bifurcation equations for f ( x , A) = 0

at (x,, AJ. The most important situation in applications is the case index f,(x,, A,) = 0 so that the bifurcation equations consist of N equations in N unknowns with N = dim Ker f x ( x o ,A,). Even in this case, the bifurcation equations do not readily yield general specific answers to the questions raised above unless N = 1. To demonstrate the utility of the bifurcation equations, we now discuss this case. Thus we suppose f ( x , A) is a C 2 function of its arguments with f(0,A) = 0 in a neighborhood of the point (0, Ao), where f,(O, A,) is a linear Fredholm operator of index zero with dim kerf,(O, A,) = 1. Then we prove Blfurcatlon Theorem for Simple Multiplicity Suppose the above hypotheses hold and moreover that fh(O, A,)z n Range f,(O, A,) = ( 0 ) for z E Ker L. Then (0, A,) is a point of bifurcation for the equation f ( x , A) = 0 and there is exactly one continuous curve of nontrivial solutions ( x ( E ) ,A(€)) bifurcating from (0, A,). (4.1.12)

Proof: The left-hand side of the bifurcation equations (4.1.8) in the present case can be regarded as a single real-valued function. Indeed, let p be a nonzero bounded linear functional that vanishes off cokerf,(O, A,,), then the bifurcation equations (4.1.8) can be rewritten

(4.1.13)

F ( P ,A)

pf(p

+ g(p, A), A)

= 0.

156

4


By a further translation of origin, if necessary, we may also suppose A, = 0. Then under the hypotheses of the theorem, we shall show that (0, 0) is a nondegenerate critical point of the C2 real-valued function F(p, A) regarded as a function of the two real variables (p, A). Consequently, by Morse:s lemma (1.6.1 l), after an appropriate change of coordinates (p, 6) +(!, A), the solutions of (4.1.13) near (0, 0) satisfy the equation - A2 = 0, i.e., the solutions (p, A) of (4.1.13) near (0, 0) consist of two curves intersecting at (0, 0). Since one of these curves consists of the (0, A) axis, there is exactly one curve of nontrivial solutions bifurcating from (0, A,). To verify that (0, 0) is a critical point of the real-valued function F(p, A) we show that g,(O, 0)p = 0. Indeed, differentiating the expression for F in (4.1.13) we find by the chain rule

c2

pf,(O, O ) { P

+ g&O,

0)P) = ~f~(o~o)g,(ol 0)P = 0.

This implies f x ( O , O)g,(O, 0)p E Y , = complement of cokerf,(O, 0). Now g,(O, 0)p E Y,, while f,(O, 0) is an isomorphism on Y , . Thus g,(O, 0 ) p = 0, as required. Consequently, F,(O, 0) = 0 and similarly FA(O,0) = 0, so that (0, 0) is indeed a critical point of F. Finally, we verify that (0, 0) is a nondegerate critical point of F with Morse index one. A simple computation, using the results of the above paragraph, shows that the Hessian matrix H,(O, 0) of F evaluated at (0, 0) is a 2 X 2 matrix whose entries are precisely the second derivatives of pf(p, A) evaluated at (0, 0) (i,e., the contributions due to the term g(p, A) vanish). Since f(0, A) = 0 near A,, we find pfAA(O, 0) = 0; while pfxA(O, 0) = pfb(O, 0) # 0 since we have assumed fhr(O, Ao)p E cokerf,(O, A,) is nonzero for p # 0. Thus the Hessian matrix H,(O, 0) is nonsingular and the associated quadratic form is indefinite. Consequently, (0, 0) is a nondegenerate critical point of Morse index one, as desired. Thus completes the proof of the theorem. Remark on slmple multlpllclty: To understand the import of the hypotheses of (4.1.12). suppose the equation f ( x , A) = 0 can be written in the form (4.1.14)

( I - AL)x

+ T(x,A) = 0,

where T ( x , A) is a C 2 function of (x, A) such that T(0, A) P E Ker(I - A,L),

fhr(O, A,)z

= - Lz = -A

-

T ( P , X’>Y)II

+ IlX’ll)llx

M{IIXIIZ +

(b) IIT(P9 % Y ) - T(,8’, .?V>ll

- x’IL

IlvllHP - P’L

MIIYII,

IIT(P0, 0,Y)II

(c)

M{IIXII

where M is a constant depending only on F. Then for sufficiently small and fixedy, the iteration scheme exists for each N and converges y)). Moreover, ?(c, y ) and y ) are continuous in c and to (K, (c, y ) , y , satisfy the equation (4.1.19), and I I X ( E , ~ )- cuoll < O(1c12),11p ( c , y ) Poll = O ( l 4 Proot: For fixed c andy, consider the real-valued function a( P, u) and the operator T( B, u)

F(E,

F(E,

defined successively by the formulas

P, cuo + u , y ) ,

(4.1.21)

T ( P, u) =

(4.1.22)

a ( P, 0) = Po + c - ‘PF( P, cu0 + T ( P, u), y ) ,

(PI

-

L)-IP*F(

where P is the projection of X + ( u o ) . We shall determine positive numbers K and to such that the mapping ( T ( P , u), a ( P , u)) exists and defines a contraction map of the set SK,< = ( ( B , u) I - Pol < K(c1, lloll < K’c) into itself for IcI < co with respect to the norm on SK,e defined by III(P, o)lll = 1 PI + \lull. Then the contraction mapping theorem implies that converges to a unique fixed point in the sequence (&+ ,,u I>_=(a(PN,uN), T( &,),u sK,. which we denote by ( P ( c , y ) , a(c,y)). Clearly this point satisfies (4.1.19) with x = cuo + B. We also prove the convergence is uniform in C. for IcI s c0, so that B(c,y) and ~ ( c , yare ) continuous functions of c since 3/, and uN are. To carry through the proof, we first note that for any fixed number K and c sufficiently small, ( B , u)cS,, (,and the hypotheses of the theorem imply that 11 T( P, cuo + u, c4)11 G K , c 2 , where K , is a constant independent of P, o E SK,ll+ I1T( Bm 0 , y )

G

M(kI + 11u11)2+ MllYll(P - Po) + Mllyll

< (2M + CM)c2.

’

’

Furthermore, for any two invertible linear operators l?,, t2 we have t; - l?? = ?l; I(l?, and since l?, = POI - L is invertible on Range(P,I - L) so is l?, = P I - L for ( P , u) E SK, with c sufficiently small, and in fact we may assume

E2)t;’,

II(PI - L ) - ?

21KPOI

-

L)-lll.

Now we determine the sphere SK, such that C3 ( P, u) = ( T ( P, u), a( P, u)) maps S , -+ SK,(. Let IlPll, IIP*I( be denoted c p , cp., respectively, then for c sufficiently small ~

(4.1.23)

IIT(b u)ll

G

~

ll(PI - L)-‘ll IIP*ll IIF(P, ~ U +O u,y)ll

s 2l1( Pol

-

L)-lllcp.(2M

+ MC)2

160

4

(4.1.24)


1))

- Pol c c-~IIPIIIlF(k €no+ T ( P , u),y)II

< c-IcpM(e

+ IIT(P, 0)1 )~
11

+ ll(P’I-L)-lll IIP*ll I I T ( P , ~ ~ o + u ’ 3 ~ 4 ) - T( P’, cw0 + 0,+,I1

< ~ K I I ( P ,-I L)-’II* + 2MII(PoI

-

L)-’II(1 + K)211YllCp*C21P- P’I.

Combining (4.1.27) and (4.1.28) above, we obtain (4.1.25). To prove (4.1.26), we use (4.1.27) and estimate differences exactly as above, so that the details may be omitted. To complete the proof we show that the sequence ( PN+ y), u N + y)) = ( a (&, q,), T( PN, uN)) converges uniformly for IcI sufficiently small, so that X) will depend continuously on c. Indeed, setting (PN+,(c,y),uN+,(c,y)) = S ( P N , uN), a straightforward induction shows that

(n,

III2‘ ( 4, ui) - Q ( P, -

12

oi-

1)111 < ( f )‘I11 @( Po, 0) - ( P o , 0)lII < 2Kco( 4 1’.

Hence for any integers n, m,

i- n

so that as m , n + 00, the above term tends to zero. Hence we conclude that ( 8 ( PN, u N ) ) is a uniformly convergent sequence of bounded continuous functions on Iw’ X X, and hence the limit function (p(c,y), a(r,y)) is continuous in c with respect to the appropriate norm. A similar argument yields continuity in y.

4.1



161

4.1 E The case of higher multiplicity

We now turn to a more difficult bifurcation problem, in which dim Kerf,(x,, A,) > 1 and f x ( x o , A,) is a linear Fredholm operator of index zero. For simplicity, we suppose that x, = 0 and (4.1.30)

f ( x , A) = ( I

-

AL)x + T ( x , A)

with

T(0, A)

=

Tx(O,A)

= 0.

In this case, since f,(O, A) = Z - AL, the only possible points of bifurcation (0, A,) occur at real numbers A, with h i 1 in the spectrum of L. However, in this case (0, A,) need not be a point of bifurcation as simple examples show. The hypothesis that I - A,L has index zero implies, by (1.3.38), that if A;' is an eigenvalue of multiplicity N 2 1 for L , then Ker( I

-

A,L) n Range( I - A,L)

=

0

Ker( Z - A,L) @ Range( I - A,L) = X . Since dim Ker(Z - A,L) = N > 1 here, for fixed A, the bifurcation equations involve the determination of real solutions of several equations in as many real variables. Thus these equations may have real nonzero solutions and in fact the solutions of these equations depend essentially on the nature of the higher order term T ( x , A). We shall discuss briefly those results that can be proven independently of any assumptions on T ( x , A) or the parity of N . We first show that if ( x , A) is a solution of f ( x , A) = 0 near (0, A,), then A can be uniquely determined in terms of the component of x on Ker(I A,L). More precisely, we prove (4.1.31) Theorem Let (x, A) be a solution of f ( x , A) = 0 sufficiently near (0, A,). If x = u + o with u E Ker(I - A,L), u E Range(Z - A,L), then there is a unique function A = g ( u ) such that: (1) f(u + 0, g(u)) = 0, (ii) g ( u ) is C 1 in a deleted neighborhood of u = 0, (iii) if f ( u u', A') = 0 and f ( u + u, A) = 0, then A = A' and 0 = u'. Proot By virtue of (4.1.8), with P = h - ' and u E Ker( &I - L), the bifurcation equations

+

can be written as

(*I

( P I - L)u + P T ( u + g ( u , P), P ) = 0.

Here g(u, p ) is the function defined in (4.1.5). Let [ u , a] denote any linear product on Ker( POI - L), then taking the inner product of (*) with u, we find (0.)

p - Po + [ u , ulr"PT(u + g(u, PI, PI, .I

= 0.

Let the left-hand side of this equation be denoted F(u, 8). Then we shall show that Fp(u,p ) # 0 in a small neighborhood of (0, Po), so that by the one-dimensional implicit

162


function theorem, there is a uniquely defined function /3 = g(u) defined near (0, Po) that satisfies (**). Indeed, Fp(u, P ) = 1 + [ u , uI-’[PT,(u =

I+o(l)

as

+ g(u, P I , P)gp + PTg(u + g(u, PI, PI, u ]

llull+Ip-pol+O.

Thus the conclusions (i) and (ii) of the theorem follow immediately. To prove (iii), note that by the above, if (x, A) and (x,, A,) satisfy (*) near (0, Ao), with x = u +!(A, u ) and x , = u + !(A,, u), then (A-’, u ) and (A;,, u ) satisfy (**). Consequently, A = A,, and so u = u’ by (4.1.5).

Definite results on bifurcation in the higher dimensional case are important for applications to concrete problems in analysis. Thus our proof of the Liapunov criterion for the periodic solutions of the system (4.1.3) shows that if the “Liapunov irrationality conditions” are violated, bifurcation results on higher multiplicity can be used in studying the preservation of normal modes under nonlinear perturbations. On the other hand, our discussion of the system (1.2.9) given in Chapter 1 illustrates the difficulty of making general statements about the existence of bifurcation points in the case of even multiplicity. Indeed, in the sequel we shall show that qualitatiueproperties of the higher order part of f ( x , A) often play a crucial role in answering such general bifurcation questions. As a second application of the bifurcation equations (4.1.8), we consider the problem of determining sub- and supercritical bifurcation of solutions for the operator equation

+

(4.1.32) x = h(Lx N x ) defined on a real Hilbert space H,at the point (0, A,). Here we suppose that llNxll = o(llxll) as I(xll+O and that N x = Bx R x , where B is a homogeneous operator of order p in x with ( B x , x) # 0 for x # 0, while IIR(x)ll = o(llxllp) as x+O. If (0, A,) is a point of bifurcation for (4.1.32) and (4.1.32) admits nontrivial solutions (x, A) in an arbitrarily small neighborhood of A, with A < A, (A > A,), then the family (x, A) is called a subcritical (supercritical) family of solutions. (See Figure 4.3) Clearly, it is important to determine sub- or supercritical bifurcation from an inspection of the structure of (4.1.32) but independently of multiplicity considerations. This can be accomplished in a large variety of cases by means of

+

(4.1.33) Theorem If in addition to the above hypotheses, the range of = Ker(Z - A,L) €€3 Range(Z - A,L), then any family of solutions of (4.1.32) bifurcating from (0, A,) will be subcritical if ( B x , x ) > 0 and supercritical if ( B x , x) < 0.

Z - A,L is closed, dim Ker(1 - A,L) < co and H

Proot: Again the result follows from the bifurcation equation (4.1.8) and the estimate of (4.1.9). The key point is that under the given hypothesis knowledge of the actual solution of the bifurcation equations is unnecessary. Indeed, using the notation of (4.1.5), the bifurcation equations (4.1.8) can be written in the form (4.1.34)

P [ ( A - h,)Lx

+ XNx] = 0,

4.2 TRANSCENDENTAL METHODS IN BIFURCATION THEORY

163

FIG. 4.3 Illustration of subcritical and supercritical bifurcation with a bifurcation diagram. where P is the projection of H + Ker(1- h,L) since under the given hypotheses Ker(Z - X,L) = coker(1- XoL). Thus, if x = p I + g with p I E Ker(f - A&), g E Range(1- X,L), then (4.1.34) becomes [(A, - X)/M,lPl = P [ B ( P , + g) + R(P1 + g)l. Consequently, taking inner products with p I and using (4.1.9), we find

+ O(llPIII)). Setting p I = lzlp with ljpll = 1, and suppose (Bp, p) > 0 so that inf(Ep, p) over all p E Ker(1 h,L) with lIpII = I is greater than some a > 0, we find that (A, - X)/Mo z I z I p - l a (1 + O(lC1)). (A, - ~ ) / ~ d l l P I I I=(BPI, * Pl)(l

Thus for X near A,, the result follows immediately from this last equation for the case of subcritical bifurcation. A similar argument if ( B p , p) < 0 yields the result on supercritical bifurcation.

4.2 Transcendental Methods In Bifurcation Theory 4.2A

Heuristics

It is an interesting fact that significant results in previously unsolved bifurcation problems have been achieved by utilizing results from topology, complex analysis, and critical point theory. This is especially true in the difficult “degenerate” cases (i.e., higher multiplicity) mentioned in Section 4.1E. In this section, we shall explore this topic relative to the equation (4.2.1) f ( ~A), = ( I - AL)x + T ( x , A) = 0, where the operators I - AL and T ( x , A) satisfy the hypotheses of Section

164

4


4.1, i.e., Z - AL is a Fredholm operator of index zero, while T(0, A) = T,(O,A) = 0 so T ( x , A) is of higher order in x. Roughly speaking, the success of these so-called transcendental methods is based on either a qualitative analysis of the higher order term T ( x , A) in the equation (4.2.1) or parity considerations of the multiplicity of the derivative f’(0, A,) at a critical point (0, A,) of f ( x , A). One attempts to distinguish an appropriate “invariant” I, of a numerical or algebraic nature, for a given operator f(x, A). The invariant ,Z is required to have the following properties: (1) Zr is a measure of the zeros of the operator f ( x , A); (2) , Z is stable under “small” suitably restricted perturbations of f ( x , A); (3) ,Z can be approximated by linearization, i.e., by means of the Frechet derivative fx off. We now consider some examples of such invariants and their role in bifurcation theory. We shall use the results of Section 4.1 to reduce the bifurcation problems to finite-dimensional considerations. Thus (apart from Section 4.2C) we shall need only those (topological) invariants ,Z for mappings f between finite-dimensional spaces, and we reserve for Part I11 those results that can be obtained by strictly infinite-dimensional arguments. More precisely, the reduction to the finite-dimensional problem is based on decomposing (4.2.1) relative to the direct sum decomposition X = Ker(Z - A,L) @ Range(Z - h,L) mentioned in Section 4.1. If P denotes the canonical projection of X onto Ker(Z - A,L) and g(u, A) denotes the function Ker(Z - A,Z) X R’ -+ Range(Z - A,L) defined by (4.1.8), then as mentioned in (4.1.9, the solution of f ( x , A) = 0 near a point (0, A,) at which f,(O, A,) is not invertible are in one-to-one correspondence with the solutions of the equation (4.2.2) u E Ker(1- A,L). ( I - AL)u + PT( u + g(u, A), A) = 0, 4.28

Brouwer degree in bifurcation theory

The degree d(f,p , D ) of a continuously differentiable mapping f,with domain a bounded domain D c R N and range f ( D ) = RN,was defined in Section 1.6C and is an invariant of the type we are seeking. Recall that the degree is an integer (positive, negative, or zero) that measures the “algebraic” number of solutions of f ( x ) = p in D , provided that f ( x ) # p on aD. First, we shall indicate the sense in which the degree d(f,p , D ) satisfies properties (1)-(3) of Section 4.2A, and thus qualifies as an integer invariant


165

I,. By virtue of its definition in Section 1.6A, d ( f , p , D ) measures the number of solutions of f ( x ) = p in D by counting “nondegenerate” solutions of f ( x ) = p with a + or - depending on the orientation preserving or reversing properties of f at the solution. Here, the solutions of f ( x ) = p in D are nondegenerate if the Jacobian determinant is not zero at each solution of f ( x ) = p in D.Furthermore, d(f,p , D )is a homotopy invariant and thus stable under small perturbations in the sense that, if f ( x , t ) is a continuous map of 5 X [0, l]-+RN satisfying f ( x , t ) # p on aD, then d ( f ( x , t ) , p , 0 ) is defined and independent of t E [0, 11. Finally, in the nondegenerate case, sgn det IJf(x)I.

d ( f , p , 0) = K X )

=P

so that the derivative off determines the “algebraic” count of the solutions of f ( x ) = p in D . We now use the fact that d ( f , p , 0 ) is an invariant to prove

(4.2.3) Theorem Suppose that the operator f ( x . A) defined in (4.2.1) satisfies the following hypotheses: (i) (ii) (iii) (iv)

(Z - A,L) is a Fredholm operator of index zero. dim Ker(1 - A,L) is odd. Ker(Z - A,L) n Range(1- A,L) = 0. T ( x , A) is a C 1 mapping with T(0, A)

= T,(O, A) = 0.

Then (0, A,) is a point of bifurcation for the equation f ( x , A) = 0. Proof: We give a proof by contradiction, by supposing that (0, A,) is not a point of bifurcation for (4.2.1). Consequently, (0, A,) is not a point of bifurcation for the equation (4.2.2). Let h(u, A) = ( I - AL)u + P T ( u + g(u, A), A) and N = dim Ker(Z - A,L). Then for ( u , A) in a small spherical neighborhood D of (0, A,), h ( u , A) # 0 on aD, so d ( h ( u , A), 0, D ) is well defined. In fact, d ( h ( u , A), 0, D ) is necessarily a constant independent of A, by virtue of the homotopy invariance of the degree function. On the other hand, for A in a sufficiently small deleted neighborhood of A,, since Ih(u, A) - ( I - AL)ul is small on aD, d ( h ( u , A), 0, D )= d ( 1 - AL, 0, D ) . A simple calculation of det IZ - ALI on Ker(1 - h,L) now shows that for A < A,, N

d(z - AL, 0 , D ) = sgn

IT ( I - AA;

I)

> 0;

I = ]

while for A

> A,,

since dim Ker(2 - A,L) = N is odd, N

d(Z - AL, 0, D ) = s g n n (1

-A&’)

< 0.

i= I

Thus d(u, A), 0, D )is not constant for A in any small neighborhood of A,;

166

4


and we obtain the desired contradiction since the given c > 0, h [ u , t(A, (1 - t)(A, E)] = 0 must have a solution on (\lull = p} for every p > 0 sufficiently small. Actually a slightly more general result involving the multiplicity of the eigenvalue A, for L instead of dim Ker(Z - A,L) can be proved. This result can be stated. E)

+

+

(4.2.3') The conclusion of Theorem (4.2.3) is valid if hypotheses (ii), (iii) are replaced by the assumption that the multiplicity of A, relative to L is odd.

Proof: The result follows by decomposing the equation f ( x , A) = 0 defined on X by two equations as in the proof of (4.2.3) but in this case we use the decomposition X = U . Ker(I - A,L)j d3 X I . By assumption . . dim UjKer(I - A,L)j (the multiplicity of A,) is odd, while I - h,L is invertible when restricted to X , . Consequently, an argument similar to the proof of (4.2.3) (utilizing the Brouwer degree) yields the desired result. The results just obtained can be considerably sharpened by placing qualitative restrictions on the higher order term T ( x , A). Indeed, suppose that for each A near A,, T ( x , A) is a complex analytic mapping of a neighborhood U of (0, A,) of X in the sense of Section 2.3. In this case we prove the following (4.2.4) Theorem Suppose the operator f ( x , A) defined in (4.2.1) is complex analytic for fixed real A near A,, and in addition (i) I - A,L is a Fredholm operator of index zero; (ii) dim Ker(I - A,L) > 0, while Ker(I - A,L)

n Range(I - A,L)

= {O}.

Then (0, A,) is a point of bifurcation for the equation f ( x , A) = 0. In fact, there is an analytic curve

c 00

X(€)

=

n= 1

an€n,

A(€) = A ,

+

c /In€", W

n=

I

of nontrivial solutions of f ( x , A) = 0 branching from (0, A,). Proof: Following the proof of (4.2.3), suppose (0, A,) is not a point of bifurcation relative to the equation f ( x , A) = 0. Then (0, A,) is not a point of bifurcation for the equation h(u, A) = 0. Now Ker(1- A,L) has even dimension. Furthermore, the map h ( u , A) = ( I - A,L)u + P T ( u + g(u, A), A) is complex analytic in u for fixed A since both T ( x , A) and g(u, A) are. The complex analyticity of g(u, A) follows from 3.3.2 since it was defined by means of the implicit function theorem. Again (by the homotopy invariance of degree (1.6.3)) for A in a small deleted neigh-


167

borhood of [0, A,] and U a small spherical neighborhood of 0 in Ker(1A,L), d ( h ( u , A), 0, U ) = d(Z - AL, 0, U ) = 1. On the other hand, at A = A,, d ( h ( u , A), 0, U ) is defined since h ( u , A) = 0 is assumed not to have solutions on the boundary of U . But at A = A,, the Jacobian determinant of h(0, A,), det(1- A,L), is zero so that the fundamental result on complex analytic mappings mentioned in (1.6.3) implies that h ( u , A,) is not one-toone in U . Hence d ( h ( u , A,), 0, U ) > 2. Thus the function d(h(u, A), 0, U) is discontinuous across A = A,, and as a consequence of the homotopy invariance of degree, this implies that h ( u , A) = 0 has a solution on the boundary of U for A in some small interval about A,. This is the desired contradiction. To prove the existence of an analytic curve of solutions branching from (0, A), we note that since (0, A,) is a point of bifurcation from the equation h(u, A) = 0, the point (0, A,) is not an isolated point of the variety V = { ( u , A) I h ( u , A) = 0 ) . Since h ( u , A) is complex analytic in u and A, V can be regarded as an analytic set near (0, A,). Thus V must contain an analytic curve (cf. 3.3.9). W

#(€)

A(€) = A ,

a,€",

=

+

m

P,€", n= 1

n= I

+

Consequently, x(c) = U ( E ) g ( u ( c ) , A(€)) can also be written as x(c) , ( Y ~ E " , and so the theorem is proven.

=z,"=

4.2C Elementary critical point theory

Letf(x) = grad F ( x ) be a gradient operator defined on the ball llxll < R in a Hilbert space H . Recall from Section 3.2 that a critical point X of F ( x ) restricted to the sphere ilxll = R satisfies the equation A,x A,f(x) = 0, where A, and A, are real numbers (with A, # 0). The associated critical value is the real number c = F(X). It is an interesting fact that in several important instances, certain critical values are numerical invariants of the type mentioned in Section 4.2A. The simplest example of such an invariant critical value is the supremum of a C z weakly sequentially continuous functional F(x), which near the origin has the form F ( x ) = + ( A x , x) 0(11x11~) when restricted to the small sphere aZ, = { x I llxll = c ) . The linear operator A is compact, self-adjoint, and we shall suppose that A has a largest positive eigenvalue A,. We now show that the number, sup F ( x ) over a&, has properties (1H3)of an invariant (listed in Section 4.2A). First, if (Y = sup F ( x ) over a Z, is attained by X E a Z, X will be a nontrivial solution of the equation g ( x , p ) = p x - grad F ( x ) for some real number p, which we shall see must

+

+

168


lie in a small neighborhood of A,. This is the sense in which this critical value measures the solutions of the equation g(x, A) = 0. In order to show that a is stable under suitably restricted small perturbations, we note that if F ( x ) is perturbed by the addition of a higher term G ( x ) = ~~(11x11~) near x = 0, then

I sup[F(x) + +)I-

aI = o(llxl12).

a z, In the same way, a can be calculated approximately by linearization since by the variational characterization of A , (mentioned in (1.3.40)), €’A, = supaX:,(Ax, x) so that (4.2.5)

-

4 e2A,I = ( SUP { 4 ( A X ,X) + ~(llxll’)) - f ~ ’ A l 1 = o(E’).

a 2, We now indicate the importance of this simple invariant in bifurcation theory. Suppose that the operator equation (4.3.1) defined in the neighborhood of the origin in a Hilbert space H can be written (4.2.6)

f ( ~ A), = x

-

A{ LX + T ( x ) }= 0.

where L is a compact self-adjoint operator and T ( x ) = grad T(x) is a higher order, completely continuous, operator with T ( x ) = o(llxll). Then we prove

(4.2.7) Theorem Let A, be the largest strictly positive eigenvalue of L. Then under the above hypotheses, (0, l / A , ) is a point of bifurcation for (4.2.6). Proof: We shall demonstrate that every sufficiently small spherical neighborhood U , = { x (1x11 < E } contains a nontrivial solution (x(E), A(€)) of (4.2.6) with IIx(e)ll = E and \A(€) - l / A , l = o(1). To this end, we make use of the facts concerning a, = sup{ t ( L x , x) + 5 ( x ) } over 32, = { x I (Ix(I= E } just described above. Assuming, for the moment, that a, is attained by an element X(E) of au,, we show that the equation satisfied by x(E), namely, p,x = Lx T ( x ) , is such that 1 - A,( = o(1). Once this estimate is obtained as c-0, (A(€) - l/A,l = ll/p, - l / A , l = 4 1 )

I

+

as required. To this end, taking the inner product of (4.2.6) with x, we find that, on using (2.5.6), P, = ll~,Il-’{(Lx€9

4)+ (Tx,,

.,>I

=2e-’{($(Lx,,x,)+ S(x,))}

+ E-’{(Tx,,x,)-2S(X,)}


I69

Thus by (4.2.5) above, l p c - A , l = 2 € - * ( a c -+A,€*} + € - l o ( € ) = o(l) as c + O .

Thus there remains only to show that a, is attained on ax,. This fact is immediate from the following argument. Clearly a, < co. Thus if {x,,} is a sequence of elements on ax, with F(x,,) = f ( L x , , ,x,,) + S ( x , , ) + a E , { x , } possesses a weakly convergent subsequence with weak limit X. Since F ( x ) is continuous with respect to weak convergence, F(X) = a€. Now X E 3 2 , since otherwise llFll < E and for some t > I , t X E ax,. A simple computation then shows that F(t,Y) > F(X), contradicting the maximality of F(x). Thus the proof of the theorem is complete. In Chapter 6 , we shall show that critical values calculated by various minimax principles are numerical invariants satisfying properties (1) A,. On the other hand, (0, A,,) is an isolated degenerate critical point of H ( u , A,) and suppose at least one of its type numbers is different from zero. By the invariance properties of Morse type numbers mentioned above, since sup llH(u, A) H ( u , A,)(/ + O in the C ' sense, as (u, A)-+(O, A,), the Morse type numbers of H ( u , A) at (0, A), Mh(0), satisfy the inequalities Mh(0) > MAo(0)in the coordinatewise sense, for each A in a small neighborhood of A,. These inequalities imply that if M,JO) = (m,, m , , m,, . . . , m,), then for A < A,, m, = 1 and m, = 0 ( i = 1, . . , N ) , while for A >A,, mN = 1 and rn, = 0 ( i = 0, . . . , N - 1). Thus m, = 0 for i = 0, 1, 2, . . . , N . This is the desired contradiction. The only remaining case to consider is the possibility that the Morse type numbers MhJO) all vanish, but this case is ruled out by the relation (1.6.15) between the Brouwer degree and the Morse type numbers. Indeed, (1.6.15) implies d ( H u ( u ,A,), 0, IuI < c) = 0, but this contradicts the fact that d(H,(u, A, - 6 ) , 0 , 1uI < c) = 1 for 6 and c sufficiently small since u = 0 is a relative minimum for H ( u , A, - 6). Theorem (4.2.15) is clearly a far-reaching extension of the analogous result in (4.2.7). Thus an immediate consequence related to periodic solutions of (4.2.8) near x = 0 is the following extension of (4.2.9).

+

+

.

(4.2.16) Theorem Under the same hypotheses as (4.2.9), the system (4.2.8) has nontrivial periodic solutions x,(t) (i = 1, . . . , k ) in any sufficiently small neighborhood of x = 0, such that the period r, of x , ( t ) tends to 27r/A, as the diameter of U tends to zero. If the real-valued function F ( x ) is real analytic, then (4.2.8) will have a real analytic curve of solutions x , ( t ) with periods ri(c) satisfying m

x,(t)=

2 a,,(t)c" n= 1

and

27T

re = -

4

+ 2* & c " n=l

Proof: The result is immediate by combining the arguments of (4.2.9) with the abstract result (4.2.15) above. Remark Weinstein (1974) has recently shown that the periodic solutions referred to in (4.2.16) are all distinct. In Chapter 6 we shall show the Ljusternik-Schnirelmann theory can be used in this connection (cf. Note F at the end of this chapter.)

4.3 SPECIFIC BIFURCATION PHENOMENA

173

4.3 Specific Bifurcation Phenomena

The size of certain critical parameters often governs the behavior of many aspects of the natural world. In many such circumstances, bifurcation phenomena play an important role in understanding. In the preceding two sections we illustrated this fact in terms of periodic motion near an equilibrium point of a system of “nonlinearly perturbed” harmonic oscillators. Here we pursue this topic in different areas of mathematical analysis in order to illustrate both the importance of the bifurcation theory developed in the previous sections and the problems involved in applying this theory to specific difficult cases. For simplicity, we have chosen our illustrations from quite well-established disciplines. Yet, despite their classical nature, each topic discussed below is filled with unsolved fundamental problems whose solution requires bifurcation theory. 4.3A

Periodic motions near equilibrium points in the restricted three-body problem

The restricted three-body problem can be described as follows: Two particles P I and P , of mass ratios p and 1 - p move in circular orbits (under Newtonian attraction) around their center of mass. A third particle P, of negligible mass moves in the plane defined by the two revolving bodies. This particle P , is subject to the Newtonian attraction of P I and P,, but is assumed not to disturb the circular motion of P I and P,. The problem to be solved is to describe the motion of P, under various given initial conditions. This problem was formulated by Euler in 1772, and has been a central issue in celestial mechanics since the profound studies of Poincare. Poincart: emphasized the importance of periodic motions in the restricted problem and in fact conjectured that any solution of the restricted problem could be approximated arbitrarily closely for a given time interval by a periodic solution. Relatively simple differential equations describing the motion of P , (in a rotating coordinate system) were found by Jacobi. These equations are autonomous and can be written (in the nondimensional form) (4.3.1) x,, - 2y, = V,(X,,Y),

(4.3.2)

y,, + 2x, = U v ( x ,y ) ,

where

(4.3.3)

U ( X , y ) = 4 (x’

+ y’) + (1 - p ) { ( x - p)’ + y’} -”’

+p{(x-

l+p)’+y’}

- 1/2

.

174


The system (4.3.1)-(4.3.2) is Hamiltonian and autonomous. Its stationary points (obtained by solving the equations (U,= Uy = 0) are the simplest solutions of this system. It turns out that there are five such stationary points: three of the form ( x k ( p ) , 0) ( k = 1,2, 3) usually called L,, L,, L,, and two others L, and L, either of which forms with the points P , and P , an equilateral triangle. (See Fig. 4.4.) Thus the points L, and L, are called triangular stationary or libration points. Here we shall investigate the possibility of periodic motions in the vicinity of such stationary points, by considerations analogous to those of Sections 4.1 and 4.2. First we observe that it is easy to find linear approximations for the desired periodic solutions. Indeed, if (xo,yo) are the coordinates of a stationary point of (4.3.1)-(4.3.2), then solutions of the linearized equations about (xo,yo) are (4.3.4) It turns out that this linearized system has one family of periodic solutions if (xo,yo)is chosen to be one of the collinear points L,, L,, L,; while at L, and L , the system (4.3.4)-(4.3.5) has either two or no distinct periodic solutions depending on whether or not the mass ratio p is less than or equal to some critical number pw The problem we take up here is the validity of this linear approximation to the periodic solutions of (4.3.1)(4.3.2). More precisely, are the periodic solutions of (4.3.4) - (4.3.5)

FIG. 4.4 Equilibrium points for the restricted three-body problem.

4.3 SPECIFIC BIFURCATlON PHENOMENA

175

accurate first approximations to the periodic solutions of (4.3.1)-(4.3.2) near each of the stationary points (L,-L,)? In Sections 4,l and 4.2, we partially solved this problem for systems analogous to (4.3.1)-(4.3.2) by bifurcation theory. Indeed, there we considered this problem in terms of the preservation of normal modes under nonlinear perturbations. We intend to show that our methods can be extended to this more difficult case. In fact we prove:

(4.3.6) Theorem If the linear system (4.3.4)-(4.3.5) admits a nontrivial periodic solution relative to the points Lj ( i = 1, . . . , 5), then in a small neighborhood of Li the nonlinear system (4.3.1)-(4.3.2) admits at least one nontrivial family of periodic solutions x, ( t ) with period T € , both depending analytically on E and such that (x,(t), T 1 for X =Ao, then the argument of Theorems (4.2.15) and (4.2.7) are applicable and we find that the equation (4.3.8) always has a curve of nontrivial solutions (x(E), A(€)) depending real analytically on z and bifurcating from (0, Ao). These solutions give rise to a nontrivial family of periodic solutions of (4.3.1)-(4.3.2). (4.3.9) Corollary The system (4.3.1)-(4.3.2) has periodic solutions in any small neighborhood of L , , L,, and L, for any p > 0. However, there is a critical number po < 1 so that the system (4.3.1)-(4.3.2) admits periodic solutior. in any small neighborhood of L, and L, for p < po, but not for p > po. In fact, apart from a countably infinite number of points Z,, in (0,po), (4.3.1)-(4.3.2) possesses two distinct families of periodic solutions near L, and L,. Proot: By virtue of Theorem (4.3.6), it suffices to determine the numbers h at which fJ0, A) = I - hB - h2L has a nontrivial kernel, or equivalently, the period h of the nontrivial solutions of (4.3.4)-(4.3.5). In order to determine the periods of these linear equations, we form the characteristic equations for this system. For L,, L,, L, this equation can be written as s4 + a( p)s2 - p2( p ) = 0, where a( p ) and p( p ) are constants. Since periodic solutions correspond to purely imaginary values of s, we see that there is only one such conjugate pair for any p since p2( p ) > 0. On the other hand, for L, and L, the characteristic equation is s4 + s2 4 p(l - p) = 0, and this equation will have the desired purely imaginary complex is the smallest conjugate roots ( 5 is,,? is2) if and only if 1 > 27p(1 - p). Thus the value positive solution of the equation 1 = 27p(1 - p). Therefore if s2 > s,, then by the extended version of Liapunov’s criterion applied to the system (4.3.1)-(4.3.2) will have two families of periodic solutions with periods near 2a/s2, 2n/s,, respectively, provided s2/sI # N, an integer. Setting D = 1 - 27p(1 - p), the excluded values of p are those values for which (1 + ~ ) / ( l -6) # N 2 ( N = 1,2, . . ). Note that these values have a limit point as p + O .

+

.

Remarks on open problems: Clearly unsolved classic problems of importance (and for which our methods apply) in connection with the above results are:

(i) the continuation of the families of periodic orbits for large amplitudes,


177

(ii) the possibility of the removal of the forbidden values of p C , for the existence of two distinct families of periodic solutions near L, and L,. (See Note F at the end of the Chapter.) 4.3B

Buckling phenomena in nonlinear elasticity

Many interesting bifurcation phenomena occur in nonlinear elasticity. Perhaps the earliest was the Euler EIusticu problem mentioned in Chapter 1. Euler’s problem consisted in giving a mathematical description of the action of an axial thrust on a uniform elastic rod, and in a paper dated 1744, he reduced this problem to a description of the solutions of the following semilinear boundary value problem: W+Pw[l-W2] 3/2 -0, w(O)=w(l)=O. Euler found that the rod deflects out of its plane or “buckles” whenever P , a measure of the axial thrust, exceeds a certain number, the so-called “buckling load,” namely the smallest eigenvalue of the associated linear problem: W+Pw=O, w(O)= w(l)=O. He also showed that the quasilinear problem could be explicitly solved in terms of elliptic functions involving the parameter P. In 1910, Von Karman proposed a set of two fourth-order quasilinear partial differential equations that can be used to describe an analogous, but more difficult, two-dimensional problem: a mathematical investigation of the buckling of a thin elastic plate subjected to arbitrary forces and stresses along its boundary. In the following years a full treatment of these equations, without extra assumptions on the shape of the plate or symmetry of the buckled plate, proved to be extremely difficult due to the nature of the nonlinearity of the partial differential equations involved. Here we propose to demonstrate the applicability of our bifurcation theory developed in Sections 4.1 and 4.2 to the mathematical investigation of elastic buckling both of plates and the more general thin curved elastic shells. The formulation of the von Karman equations is as follows: We consider a thin elastic body B that is flat in its undeformed state subjected to a compressive force (of magnitude A) acting on the boundary of B. Then the stresses produced in B , as measured by the Airy stress function, f ( x , y ) + AF,,(x,y) and the displacement of B from its flat state u ( x , y ) are defined by the following quasilinear elliptic system (cf.( 1.1.12)) (4.3.10)

Azf= - 4 [ u , u],

A2u = AIFo,U ] + [f,ul,

178


where A’ denotes the biharmonic operator and

[ f , g l = f x x g y y + f,gxx

- 2fxygxy’

If we represent B as a bounded domain G in R 2 and the boundary of B as aG, we may consider the following boundary conditions associated with (4.3.10): u = u, = uY = O (4.3.11) f=f,=f,=O

on

aG

Here Fo(x,y) is the function obtained by solving an associated inhomogeneous linear problem, and is a measure of the stress produced in the undeflected plate if it were prevented from deflecting. The resulting equilibrium states are called “buckled” states, and the problem is referred to as “elastic buckling.” In order to study the onset of buckling, it is generally supposed that the nonlinear terms [u,u] and [ f , u ] in the equations (4.3.10) can be neglected. Thus the classical linearized problem of the buckling of plates is known to be described by the following linear eigenvalue problem: (4.3.12)

A% - A[ Fa,w ] = 0

in 0, on a0.

W = W ~ = W= O (4.3.13) Y In terms of our formulation via functional analysis the linearized problem consists in studying the selfadjoint linear eigenvalue problem

(4.3.14) , w = ALw in W2,,(52). Throughout our work we shall impose the following condition on the function Fo(x, y): The operator Lw = [Fa,w ] is a bounded compact operator on W2,2(L?).The validity of this assumption will be assured, for example, if all second derivatives of F, are uniformly bounded in 52. Furthermore, we remark that this assumption allows us to consider operators L with positive and negative eigenvalues. This situation corresponds to the physical action of compression on one part of 3 0 and tension applied to another part of 8 0 . Indeed the spectrum of (4.3.14) consists of eigenvalues {A,,} forming a sequence of discrete numbers tending to + 00, -00, or both. The multiplicity of each A,, is finite, and zero is not an eigenvalue of (4.3.14). We mention a simple example of linearized problems for clamped plates. Example A circular clamped pIate subjected to a unform compressive pressure at its edge Here the equations (4.3.12)-(4.3.13) reduce to (4.3.15) Ah50 - X AW = 0, w = W, = w = 0.

The study of the solutions of this problem then becomes an analysis of radially symmetric eigenfunctions and nonradially symmetric eigenfunctions.

179


The radially symmetric solutions w = w(r) of (4.3.15) can be explicitly determined by the zeros of the Bessel function J , ( r ) . These eigenfunctions are simple and can be characterized as solutions of the second order ordinary differential equation r'4 + r $

+ ( r 2 - I)+

=0

which are finite at r = 0. Nonradially symmetric eigenfunctions can be obtained in the form w(r, 0 ) = R ( r )

{

sin pll cospll'

These eigenfunctions are not necessarily simple, but the first eigenfunction is known to be axially symmetric, and simple.

Now we indicate in just what sense (i) the eigenvalues A,, and eigenfunctions are valid first approximations to the solutions of (4.3.12)-(4.3.13) and (ii) the relevance of eigenvalues A,, for the understanding of buckling phenomena. To this end, we prove (4.3.16) Theorem (i) Suppose A,, is an eigenvalue of the linear system (4.3.14). Then (0, A,,) is a point of bifurcation for the nonlinear system (4.3.12)-(4.3.13). Thus for each A,, there is a one-parameter family of solutions of (4.3.12)-(4.3.13), (w,,fL, A,) depending analytically on E such that

z m

m

wc=

wn€n.

=

n= I

f,,€",

A, =A,,

+

n=2

z m

&€".

n=

I

and w I is a solution of (4.3.13)-(4.3.14). (ii) The system (4.3.12)-(4.3.13) has no solutions in the interval (A- I , Al), where A , and A - are the smallest positive and negative eigenvalues of (4.3.13)-(4.3.14), respectively. (iii) Near w = 0, A = A,, the nonlinear system (4.3.12)-(4.3.13) has no solutions for A < A, ( N = 1, 2, . . . ) for A, > 0 and no solutions for A > A, for A, < 0.

,

Prod of (I): In Section 2SC, we showed that the solutions of the system (4.3.12H4.3.13) can be put in one-to-one correspondence with the solutions of the operator equation

(4.3.17)

w

+ c w = ALW

defined on the Hilbert space H = W2.2(Q), where the operator C(w) is defined by first defining the bilinear operator C(w,u): H x H + H implicitly by

(4.3.18)

(C(w, u), 'p) = k [ w ,u ] q ,

gwC~(s2).

Then C(w) = C(w,C(w, w ) ) is a completely continuous gradient operator, homogeneous of degree 3, and such that (Cw,w) > 0. Thus setting f(w, A) = w Cw - U w , we note that by virtue of Theorem (4.2.15), all the points (0,A,) at which dim Ker f,(O, A,) > 0 are points of bifurcation of (4.3.17) and these associated numbers A, coincide with the eigenvalues A, of (4.3.12H4.3.13).Sincef(w, A) is real analytic in w and A, the expansion in (i) follows again as in (4.2.16).

+

180


4

Proof of (11): Suppose ( w , f ) was a solution of (4.3.12H4.3.13) for At[A-,, A,]. Then (w,A) would satisfy (4.3.17). Taking the inner product of (4.3.17) with w , and using the variational characterization of A, and A - we find that ( C w ,w ) = 0.Consequently, jn[w,w]q = 0 for all 'pfH so that w satisfies the system

,,

(4.3.19)

[ w ,w ] = w,,wYy - w$ = 0

(4.3.20)

D%l,, = 0,

Thus the surface w

in 51

1 1 < 1.

has zero Gaussian curvature and is therefore developable. This surface is covered by straight lines and by virtue of (4.3.20), w ( x , y ) = 0 in D. = w(x,y)

Proof of (111): The fact that ( C w ,w ) > 0 for any nontrivial solution of (4.3.12H4.3.13) just proved above, and an application of (4.1.33) to the equation (4.3.17) yields the desired result.

Remark on a sharpenlng of (4.3.16): In Chapter 6 we shall establish an important sharpening of the result (i) above, viz., if A,, is an eigenvalue of multiplicity k , then, roughly speaking, the nonlinear system (4.3.12)(4.3.13) has at least k one-parameter families of solutions bifurcating from (0, 0, A,,). (For a more precise result, see Section 6.7C.) The buckling phenomena associated with thin elastic shells (i.e., elastic structures that are initially curved) are considerably more complex than in the case of elastic plates, despite the similarity in the analogous von Karman equations. Indeed, it is a well-known experimental fact that linearization often does not explain the obserued deformafions. In order to illustrate this fact, we consider a thin shallow shell S of arbitrary shape whose plane projection is a bounded domain D in the xy plane with boundary aQ. Suppose the shell is acted on by an external force Z ( x , y ) and by forces along an. Then, subject to appropriate boundary conditions, the equilibrium states of S will be determined by solving the following von Ktirman equations:

- f [ w ,w ] - (k,~,),- ( k z ~ ~ ) ~ , h = [ f , wl + ( k , f , ) , + ( k l f y l y + Z.

(4.3.21)

A?=

(4.3.22)

A

Here k, and k , denote the initial curvatures of the shell in cross sections parallel to the zx and zy planes, respectively. Thus the effect of initial curvature on the von Karman equations is merely the addition of linear curvature terms in ( w , f l . Moreover if Z = A+, is chosen so that together with the boundary conditions

(4.3.23) (4.3.24)

w=w =w X

f,,,=Aq,

Y

=o

fi, =A*,

on

as2

the system (4.3.21H4.3.24) has a solution ( w , n = (0, AF,) for all A, (i.e., a solution depending linearly on A, in which the midsurface of the deformed shell is stretched but not bent). Here n and T represent derivatives in the normal and tangential directions, respectively; 8 , and 8, represent edge stresses applied on aa; and A measures the magnitude of the edge stresses. We note that such a function Z = A q o depending linearly on A can always be determined for given smooth q,,qZ.Indeed if AF, is the solution of the Dirichlet problem A2F = 0 together with the boundary condition (4.3.24) we compute qoby the formula (klFox)x - (k2FOJy = -8,,. Hence w = 0, F = AF,, satisfies the resulting equations (4.3.21)-(4.3.24). Now writing a tentative solution of the full system (4.3.21)-(4.3.4) as w = w , f = F + AF, we find that the following system of equations determines the desired equilibrium states:

(4.3.25)

A ~ F ,- I2 [

(4.3.26)

A h = [ F , W ] + A[Fo, W I + (kiF),

9

WI

- (k,w,), - (k2WJy? -I-(k2F)y-


(4.3.27)

w=w =w X

181

=o

Y

on a 0 .

F = F X = FY = O

The associated linearized system in this case can be written (4.3.28) (4.3.29)

=

- ( k , w . A - (k2WY).”,

A h = A[FoWI + ( k i F ) ,

+ (k,F),

together with the boundary conditions (4.3.27). Clearly (by following the arguments of (2.5.7)) this system can be written as an operator equation in the Hilbert space H = W,, @): (4.3.30)

w

+ L:w = ALW,

where Lw is defined as in (2.5.7) and

Under the assumption I made above, the spectrum of (4.3.30) has exactly the same properties as in the plate case.

The relationship of the system (4.3.25)-(4.3.27) and its linearization about (0, A) is expressed in the following (4.3.31) Theorem (i) Let A, denote an eigenvalue of the linearized system (4.3.30). Then (0, A,) is a point of bifurcation of the system (4.3.25)-(4.3.27) and there is a one-parameter family (w!,), f!,), A!), depending analytically on c (for small c) such that w y = LW,(X)

A:,)

= A,

+ 0(€2),

f‘,”

=O(2)

+ O(€),

where w, is a normalized eigenfunction of (4.3.30). (ii) the trivial solution ( w . A) = (0, A) attains the absolute minimum of the potential energy for 0 < A c KO, where KO is the smallest positive eigenvalue of the associated linearized plate equation w = ALw. However. for A E A J , the trivial solution (0, A) (although a relative minimum) is (in general) not the solution that attains the absolute minimum of the potential energy.

(xo,

Proof of (I): Again we can reformulate the system (4.3.25H4.3.27) as an operator equation in the Hilbert space H . This operator equation can be written down as in (2.5.7), first as the pair of equations

(4.3.32)

F = - :C( w. w )

(4.3.33)

w

=

-

C ( F . w ) + ALw

LIW

+ L,F,

Then by substituting (4.3.32) into (4.3.33), and setting Cw C ( w , C ( w , w ) ) as in (2.5.7) (4.3.34)

G ( w ,A ) = w

+

C ( W )+ C ( W ,L ~ w+) 4 L , C ( W ,W ) + L ~ w - ALw = 0.

Now the points A, at which G, (0, A) is not invertible coincide with the eigenvalues of (4.3.30). To show that each such A, is a point of bifurcation independent of dim Ker(I + Lf - A,L), we show that G ( w . A) is a gradient operator and then we apply Theorem (4.2.15). A simple computation shows that if G(w, A)

= IIW’IIl+

11 ;C ( w , w ) + LiWlI2 - A(Lw, w ) ,

then g W ( w ,A) = 2 G ( w , A). so that G ( w , A) is a gradient operator. Furthermore, G ( w , A) can be put in the standard form G ( w , A) = ( I - AL)w + T ( w ) by defining a Hilbert space norm ~ ~ ~ as, new , on W,,,(Q) equivalent to the norm IIuI/,,, =[,lAw12. by setting ~ ~ w+ IILwlJ$., norm.

182


Proof of (11): The potential energy of the system can be represented by the functional 4 ( w , A) defined in the proof of (i). By the variational characterization of for Ac[O, A,], 4 ( w , A) > 1) t C(w, w) + L,w1I2 > 0. Since 8 (0, A) = 0, the trivial solution (0, A) attains the infimum of G (w, A) for hc[O,A,]. On the other hand, for hr[x,, A,] and arbitrary z E H,the quadratic form

x,

(G,,(O,

A)z, z) = (z,z>

>0

+ ( L L 2 ) - A(Lz, z) (by the variational characterization of A,).

[x,,

Thus for A E A,], the trivial solution is a relative minimum of the potential energy function G (w, A). In order to show that (in general) the trivial solution is not an absolute minimum, we note that if (W,F,A) is a solution of (4.3.24), then

4(W,

X) = - t IIC(rj, W ) ( 1 2 - f(C(W,W),LW).

+

Now if dim Ker(1 L2 - AIL,) = 1 and (C(u,, u,), Lu,)# 0 as is the case in general for u , E Ker(1- Lz - AIL,), then G (W, can be made negative for A < A,. Thus, in general, the trivial solution is not an absolute miminum for a range of values A below A,.

x)

It is important to study the stability properties of solutions of the system (4.3.10H4.3.11).In fact there are various theories to explain the physical principles by which the plate “chooses” a particular solution or “bifurcated state” among the various possibilities present at each value of A. Here we state one such principle which dates back to Dirichlet. Prlnclple of Least Potentlal Energy: At a particular value of A, the plate selects a state at which its potential energy is least. Conversely, an equilibrium state that is not a relative minimum of its potential energy is unstable.

For the present problem, the potential energy of an equilibrium state defined by u ( x , y ) is defined, up to a constant factor, by

(4.3.35) V ( u ) = ( u , u ) + f (Cu, u ) - A( Lu, u ) . Therefore, the unbuckled state of the plate uo has potential energy V(uo) = 0. (4.3.36) Theorem Any buckled state of the plate defined by u = u ( x , y ) has strictly negative potential energy, i.e., V ( u ) < 0. Consequently, the unbuckled state is unstable for A > A,. Proof:. For any buckled state, (u, u)

+ (Cu, u ) = A(Lu, u).

Thus V ( u ) = - 4 (Cu, u )

< 0,

since

u

# 0.

Hence, by virtue of the principle of least potential energy, the plate always buckles out of the plane when X > A,; the trivial solution is unstable for A > A,.

I83


4.3C Secondary steady flows for the Navier-Stokes equation

In a large number of cases the structure of the steady solutions of the Navier-Stokes equations governing the motion of a viscous incompressible fluid depends crucially on a single real dimensionless parameter R , called the Reynolds number. For sufficiently small R , there is a unique “laminar” stationary flow that satisfies the Navier-Stokes equations. In fact, in Chapter 5, it will be shown that under very general circumstances the Navier-Stokes equations always admit at least one stationary solution for any positive value of the Reynolds number R . Nonetheless, it is observed in many circumstances that these stationary solutions of the Navier-Stokes equations are unstable for large Reynolds numbers. Indeed, as the Reynolds number increases, unsteady, highly irregular, (turbulent) fluid motions are observed experimentally. The explanation for this phenomenon on the sole basis of the nonlinearity of the Navier-Stokes equations is an outstanding and unsettled problem. Here we investigate the beginning of this transaction from laminar steady flow, by means of bifurcation theory. In particular, (in certain cases) we show first that the eigenvalues of certain linear operators can be associated with points of bifurcation of the NavierStokes equations; and secondly, that at these points of bifurcation an “exchange of stability” phenomenon takes place. (I) General problem of secondary stationary flow Let f2 be a bounded domain in R” ( N = 2, 3) with boundary an. The Navier-Stokes equations for the motion of a viscous fluid flow under assigned forces f and a are p AU = ( u . grad)u + V P + f , (4.3.37) div u = 0, u I aa = a. Here the vector u and the scalar P ( x ) are unknown. We suppose that f - vfo depends linearly on a parameter v and that the system (4.3.37) admits a known solution of the form ~(v= ) vuo, P ( v ) = Po(xo,v) for all real v. (4.3.38)

+

In this case we seek other solutions of (4.3.37) of the form L, = vw L,(v), P = cp + Po. Then to determine w and p , setting h = v / p , we investigate the nontrivial solutions of the equation (4.3.39)

Aw = h{(w. grad)u

div w = 0, w I ao = 0.

+ ( u . grad)w + (we grad)w} + Vp,

184


Clearly, by utilizing the results of Section 2.2D (cf. Note C of Chapter 2), we can reformulate the solutions of (4.3.39) as solutions of an operator equation of the form

+

f(w,A) w - A{ LW Nw} = 0 in the real Hilbert space 8, of solenoidal N-vectors obtained by completing each component of the solenoidal N-vectors in CF(Q) in the Sobolev space 6’,,$2). The operators L and N are defined implicitly by the formulas (cf. Note C of Chapter 2)

(4.3.40)

(Lwsrp)~, =

- Q{

( N w , rp)k, = -

w . grad u }

J’,{ w . grad w }

+ {uagrad w}].rp,

“p.

As mentioned in Note 2C, the Sobolev imbedding theorem implies that L and N are compact mappings of fi,-+ 8,. Actually we shall restrict consideration to those problems in which it can be established that dim Ker(Z - AL) is odd in fi,. Indeed the operator N is neither a gradient map nor complex analytic on the real Hilbert space fi,, so that the only general results applicable to (4.3.40) are those that place a restriction on the parity of dim Ker(Z - AL). In order to analyze Ker(Z AL), it will be necessary to specialize Q as well as the vectors f and a . Thus our bifurcation analysis of Sections 4.1-4.2 shows that in order to prove that the Navier-Stokes equations admit secondary stationary flows distinct from ( u ( v ) , P(v)), .it suffices to find points of bifurcation (0, A) of the equation (4.3.40). Consequently, we must determine the real eigenvalues X of the linear operator f’(0, A)w = (Z - AL)w in and determine which of these correspond to points of bifurcation of (4.3.40). A solution u(x,t ) of the time-dependent Navier-Stokes equations (1.1.18)-(1.1.19) is called stable or unstable according to whether any small perturbation of the data defining the solution u(x,t ) gives rise to a solution u(x,t ) that does or does not remain close to u(x,t)for all t , in an appropriate norm. For stationary states u ( x ) , this stability criterion can sometimes be verified by linearization in the following manner. One considers a solution of the equations (1.1.18H1.1.19) in the form u(x, t ) = e “‘w(x) + u(x) neglecting higher order terms in w(x);i.e., we consider the spectrum of the Navier-Stokes operator linearized about u ( x ) . If this operator has an eigenvalue with positive real part, then u ( x ) is called unstable according to linearized stability theory; while if all eigenvalues of the linear operator have negative real part, u(x) is called stable (in the linearized sense). Thus in order to study linearized stability theory it will be necessary to study the nontrivial solutions (w,a) of the equation

A,,

(4.3.41)

f,(u,

A)w

=

-OCW,


185

where f is the imbedding of Wl,,(a)+ L,(L?) in the Hilbert space k,,and f,,,(u, A) denotes the Frechet derivative of the operator f ( w , A) (defined in (4.3.40)) evaluated at the stationary solution ( w , A). As a first example, to illustrate the nonuniqueness phenomena just described, suppose is a smooth surface of revolution that does not contain points on the axis of revolution z. Then for this special geometry (in terms of cylindrical polar coordinates ( r , 0, z) we seek secondary, rotationally symmetric, stationary flows w = w ( r , z). We suppose that the external force F = (0, Fo(r), 0) I S such that (4.3.37) admits a trivial solution o ( r ) = (0, uu,(r), 0) for u E ( - co, co). From (4.3.40), the resulting NavierStokes equations for w = (w('), w@),w'")) in cylindrical coordinates are written in the operator form f ( w , A) = [ I - AL]w AT(w) in the closed subspace A, of H I consisting of rotationally symmetric, solenoidal vectors with inner produc!

+

(w, U ) ~ , = / ~ ( V W V c. ) r dr dz,

where D is the axial cross section of Q. Then, if w, = - o , / r + u,/r).

and

g = -(do,/&

In general, L is not a self-adjoint operator; but if oo(r) = r B with p < - 1, it can be made self-adjoint by a slight change in the inner product on A,. Indeed, we define an equivalent inner product on A , by setting

Since 2 w ( r ) = ( - 2 / (

p + I))g(r),

= [w,L'p,I,-,,

the operator L is both compact and self-adjoint. Consequently, there is a countably infinite sequence of isolated real numbers 0 < A , < A, < . . . < A, + 00 such that 0 < dim Ker(I - A,L) < co (i = 1, 2, . . . ). An immediate consequence of these facts is the nonuniqueness of stationary states for (4.3.37). (4.3.43) Theorem Let 52 be described as above. Then there are vectors f and boundary conditions a such that the corresponding rotationally symmetric stationary states of the Navier-Stokes equations (4.3.37) are not unique.

Proof: Let X = u / P be any eigenvalue A, with eigenvector u, described above. Let

0, =

186

4


4 (a + ui) and u2 = f (a - ui), where a = (0, hra, 0) and fi < - 1. Then 0, and u2 are solenoidal, and take the same value on asl. Furthermore in a, ( I - hL)u, AT(u,) = ( I - U ) u2 AT(u3. Consequently, if the vectorf is this common value, both u , and u2 are the desired rotationally symmetric stationary states. We now prove

+

+

(4.3.44) Theorem Under the above conditions, if dim Ker(I,- A,L) is odd, then the Navier-Stokes equations (4.3.37) admit a family of secondary rotationally symmetric stationary states w i= wi(r,z, c) for A(€) near A, such that

wi(r,z , 6) = cui(r, 0 ) + o(lcl*),

h(c) = h i

+ ~(lcl),

where ui is a solution of the linearized equation for A = hi.

-

Prool: The proof is immediate from (4.2.3) and the fact that in 2, the operator equation f ( w , A) ( I - hL)w + T w = 0 is such that L is compact and self-adjoint, while T is a C 1 operator of higher order near w = 0. (4.3.45)

Corollary Let A, denote the smallest positive number such that dim Ker(I - AL)

> 0. Then the trivial solution u0(v) is stable (in the linearized sense) for (hl < A, and unstable (in the linearized sense) for Ihl > A,. Prool: For 0 < h < A,, the equation ( I - AL)u = - a e u , (IuJI= 1, implies that ( e u , u)a = -((I -hLu), u). Thus by the variational characterization of A,, for u # 0,( e u , u)a < 0. Consequently, a < 0 and the stationary state associated with the trivial solution, (0, A), is linearly stable. For A > A,, the smallest eigenvalue aI of I - hL is characterized as

Consequently, the trivial solution (0, A) is unstable for A > A,, according to the linearized stability theory.

(11) Taylor vortlces The flow of a viscous incompressible fluid between two rotating concentric cylinders of infinite length is an excellent (although more complicated) example of the type of secondary stationary flows just mentioned. Suppose the radii of these cylinders are denoted R , , R , with R , < R,, and these cylinders rotate with angular velocities w , and a,, respectively. Suppose that cylindrical coordinates ( r , 0, z ) are chosen with the z axis coinciding with the common axis of the cylinder. It can then be easily shown that with f = (0, vfo, 0) and a = 0, the Navier-Stokes equations admit a solution ( u ( r , v), P ( v ) ) with u(r, v) = (0, u,?(r), 0). This solution is called Couette flow, and we shall find relations between rl, r2, a,,and w2 such that (4.3.37) admits “periodic” secondary flows of the form u = u(r, v) w ( r , z ) (i.e., axisymmetric flows) that are periodic in z . (See Fig. 4.5.) Such flows are called Taylor vortices after G. I. Taylor who discovered them experimentally in 1923, and studied them mathematically by linearizing the Navier-Stokes equation about the Couette flows.

+

Proceeding as above, we seek vectors w = (w(‘), w@), w(’))that (i) vanish for r = rI, r,, (ii) are 2 r / a 0 periodic in z , where a. is to be determined, (iii)

187


Outer cylinder

FIG. 4.5 Notation for Taylor vortices.

possess no net mass flow in the z direction, i.e., j:w(')r dr = 0, and (iv) w('), w(') are even in z , and w(@)is odd in z . Such conditions can easily be incorporated into an admissible class of solutions for (4.3.37). Clearly this class K , is a closed subspace of k,. In order to prove the existence of Taylor vortices, we prove the existence of a point of bifurcation for the operator equation (4.3.40) in il Actually one can prove (4.3.47) Theorem If w I > 0 and w2 2 0, then the Navier-Stokes equations (4.3.37) admit secondary Taylor vortices for the configuration just described above, provided o,ri < w , r t (i.e., if the inner cylinder rotates with a sufficiently large angular velocity). Proof: By what has been mentioned abpve, it suffices to analyze the spectrum of the linearized operator L in the Hilbert space K,. Thus we wish to analyze the system (4.3.48)

AU("

- u ( ' ) / r 2 - aq/ar Au")

-

u(')/r2 ~

+ Xw(r)u(') = 0, + Xg(r)u") = 0, ~

(i/r)(a/ar)(ru(r))

(

-2

aq/az )

= 0,

+ a u ( z ) / a z = 0,

+

r,:

u(')r dr =

o

where o ( r ) = a b / r 2 and g ( r ) = - 2 a . Thus we seek solutions of this system of the form u = (u('), u(@),u ( I ) ) = ( u ( r ) cos az, u ( r ) cos az; w(r) sin az). Substituting these into (4.3.48) and eliminating y we find the functions u ( r ) , u ( r ) and eigenvalues X by solving the system of ordinary differential equations ( L - a2)'u = 2a2Xo(r)u, ( L - a2)o

= -Xg(r)u,

u ( r , ) = w(rJ = u,(r,) = 0

(i

=

I, 2),

188


where L = d 2 / d r 2+ ( l / r ) d / d r - l/r2. Let G , ( r , r ’ ) and G,(r, r ’ ) be Green’s functions for the differential operators - r(L - a,) and r(L subject to the above boundary conditions. Both G I and G, are continuous and symmetric in r and r’. In fact, findidg the solutions of (4.3.48) in k , is clearly equivalent to finding the nontrivial solutions of the system (4.3.49)

P = uG&,

(4.3.50)

o = pG, gG2wo,

g ~ ,

p = 2a2A2.

Now we prove that if a number p is a simple eigenvalue of (4.3.49), then A = T@ is also a simple eigenvalue of L. Finally, we observe that Green’s functions GI and G, are “oscillatory kernels”, (cf. Karlin (1968) and thus, provided o ( r ) and g ( r ) are both positive, the operator B = G p G , g is also. Therefore, the operator equation (4.3.49) has a sequence of simple eigenvalues 0 < p l ( a ) < p2(a) < . . . < pn(a)< . . . . Thus the spectrum of I - AL consists of the eigenvalues Aik(aO)= & [ ~ ~ ( k a , , ) / f k ~ aNow 2 ] ~ /to~ .ensure that (4.3.48) has simple eigenvalues, we note that the functions pk(a) are real analytic functions of a. Setting A,(ka) = Aiik(a),the function = A,(&) - A,(sa) is real analytic in a (and is easily shown as not identically zero for i # r ) . Thus the zeros of AgjS are denumerable and so the set Z = (alA$!s 0; r, s = 1, 2, . . . ; r , s # i , k ) is denumerable. Clearly the positive numbers a in the complement of 2 generate simple eigenvalues of L, so that for such values a, dim Ker(1- A(a)L) = 1 and Ker(1- AL) n Range(I - AL) = (0).Thus (0, A(a))is a point of bifurcation of (4.3.40), in this case by (4. I. 12).

-

Before proceeding to a discussion of the occurrence of bifurcation phenomena in the theory of complex manifolds hinted at in Section 1.1, we refer the reader to the brief discussion on analysis on complex manifolds in Appendix B. 4.30 Bifurcation of complex structures on compact complex manifolds

Complex manifolds often depend critically on parameters in the sense that the complex structures associated with these manifolds change as the associated parameters, defining the structure, vary. Each member of a family of complex structures M , depending analytically on complex parameters o may be complex analytically homeomorphic, in which case we consider the family “trivial” (from the point of view of bifurcation theory). Here we consider the question of finding nontrivial deformations of a given complex structure, i.e., a family of complex structures M , depending analytically on complex parameters w, but such that the members of the family M , are not complex analytically homeomorphic. The approach to this problem via nonlinear partial differential equations was mentioned briefly in Section 1.1, and here we shall explore certain analytic aspects of this problem in more detail. (For a more complete discussion, we refer the reader to the monograph by Kodaira and Morrow (197 l).) The connection with nonlinear partial differential equations is made as follows. Let %, be a compact complex analytic manifold of complex


189

dimension N . We shall apply the bifurcation result of Section 4.1 to construct a nontrivial family !!Ktof complex structures on X0depending continuously on a finite number of complex parameters t , for It1 sufficiently small, an_d with Socorresponding to t = 0. Two complex structures 9Ro and 9 2 defined on the same manifold %-are close if in terms of the local holomorphic coordinates E ' , . . . , 5" of %, the form d t J may be expressed in terms of suitable local holomorphic coordinates zl, . . . , z" of a0 by setting n

(4.3.51)

d5J = dzJ

+ 2 v i ( z ' ,. . . , z n ) d F k , k= I

-

where T , ~is small in some common coordinate patch. The complex structure of % defines a splitting of the complex first order differential forms on % into the direct sum of a n n-dimensional subspace T and its complex conjugate space; and so defines an almost complex structure on 91L.Thus if bXis an almost complex structure on "X satisfying (4.3.51), the form w = Z w k K z k is a well-defined vector (0, 1) differential form on It is known (by the Newlander-Nirenberg theorem, cf. Note D of Chapter 3) that this almost complex structure defines a complex structure on % if and only if the following "integrability condition" is satisfied: I

.:no.

(4.3.52)

L - [a,03

= 0,

where for any vector-valued ( 0 , p ) and (0, q) forms w and a, [a, a] is a ( 0 , p q ) form defined whose i-th component is

+

[ a,

01 = 4 2 (a] A a i d + ( - l)awJ A a,d)

i where aj = a / dzj and a = p q 1. So that the bilinear operator [a, a] satisfies the following identities

+

(4.3.53)

(i)

[ u , w ] = ( - ~)'"w, a]

(ii) (iii)

$[a,

01=

[Zu, w ]

+ (-

~ ) ~ [ $a]; a,

[[a, a],a] = 0.

In this sense the determination of nontrivial families of complex manifolds can be reduced to a study of (4.3.52). Indeed, complex structures on 92 can be constructed by finding solutions of the equation (4.3.52) and, in fact, we shall find a family of solutions of (4.3.52) near w = 0. More precisely, we prove (following Kuranishi (1965)) (4.3.54) Theorem Let m be the dimension of the vector space H '(92, a),the first cohomology group on the compact complex analytic

190

4


manifold % with coefficients in the sheaf of germs of holomorphic vector fields (0).Then the equation (4.3.52) has a family of solutions near w = 0 depending on m complex parameters and lying on a complex analytic set near the origin in C'". Moreover, if the cohomology group H2(%, 0) vanishes, there is a nontrivial family of deformations for each element of HI(%, 0). Proof: The result is obtained in a sequence of steps by considering the solution of the enlarged system of partial differential equations (4.3.55) (i) Zw = [a,w ] ; (ii) ZTw = 0, together with its linearization near w = 0, (following Kuranishi)

STw = 0. Here the operator ZTis the L, adjoint of 8. (4.3.56)

(i) $a= 0;

(ii)

Now, as described in Appendix B, the solutions of (4.3.56) coincide exactly with solutions of the vector Laplace equation (4.3.57)

ow = 0,

where 0 =

a%+ 8gT.

Thus the solutions of (4.3.56) coincide with (complex) the vector-valued harmonic (0, 1) forms Ho, I ( 0 ) defined on the complex analytic manifold 'X, and these forms, in turn, correspond to the elements of HI( Gx , 0). This last conclusion follows from Hodge theory.

a,

Step 1. Reformulations: In order to apply the bifurcation theory of Section 4.1 to this problem, we first rewrite the system (4.3.55) by means of the Hodge-Kodaira decomposition theorem (of Appendix B). Using the fixed Hermitian metric on 9L0, we introduce an L, scalar product onQo,p, denoted for w , u by (a,a). Then by definition, (ab, w ) = (a, 8 a). Let H be the projection of + HO,p( %), and G o be the projection of ~ o , p ~ [ H o , p ( ' Then X ) ] *G. commutes with 3 and ZT, while if one sets Q = 8 TG, one finds that for w €Ao,p(since H and G o are complementary projections)

(4.3.58)

w = Hw

+ ZQw + QZw

or

w = Ha

+ OGo

Consequently, (4.3.55) implies that o satisfies (4.3.59)

w = Hw

+ Q[w, w].

Here we have used the fact that, since Conversely, we shall show that

STw = 0, so that

ZQw = 8GSTw = 0.

(*) if w satisfies (4.3.59) (and is sufficiently small), then w will also satisfy the system (4.3.55) on % , provided H [a,w ] = 0, where H is the projection of Ao, onto the vector-valued (0, 2) harmonic forms Ho,,( 9 2 ,0).

191


As a preliminary step in this direction we note first that if w satisfies (4.3.59), ZTw = 0 since aTQ and STH are both identically zero; and secondly, by operating on (4.3.59) with 8, and then using (4.3.58), (4.3.60)

Zw=ZQ[w,w]=OG[w,w]-a

a

-T-G

[o,o]

01 = - H [ u , 01 - Q a [ w ,

- [a.

w].

Moreover, we shall show later that H [a,w ] = 0 implies that Q $ [ w , w ] = 0. To investigate the smoothness of the solutions of (4.3.59), we consider the semilinear elliptic equation

(4.3.61)

00- aT[o,w]

=0

a,,.

defined on We begin by observing that the smooth solutions of this equation include the smooth solutions of (4.3.59) so that the regularity properties of these solutions can be deduced from the properties of (4.3.61). Indeed, if w satisfies (4.3.59), then

(4.3.62)

ow = OQ[w,a] = OGZT[w, w] = aT[w, w].

Thus we conclude that if w is any known solution of (4.3.59) in the Sobolev space Wk on k-times differentiable (0, 1) forms, the smoothness properties of w can be deduced from the regularity theory for the nonlinear strongly elliptic second order systems, provided k is chosen sufficiently large. In fact, w can be considered as a C" function. (For details, see Kodaira and Morrow (1971).) The equation (4.3.59) can now be reformulated as the following operator equation in w k ;

+

(4.3.63) Lw B ( w ) = 0 with Lw = w - Hw, B ( w ) = Q [ w , w], where the bounded bilinear operator B satisfies the following estimates for arbitrary w, 0 E H and sufficiently large k:

IIB(w)

-

B(w)l\k

c(llllwkl IIGllk)l10 - w l l k *

+

where c ( x , y ) - + O as 1x1 ly(-+O. This fact is a consequence of the following routine but lengthy estimates: II[w,

0111k-1

CIIWllkllallk;

llQw\lk+l

cllwllk,

where C is an absolute constant and k is chosen sufficiently large so that by Sobolev's inequalities (1.4.12) yields pointwise estimates. Moreover, we note that since W, is a Hilbert space over the complex numbers, the mapping B is complex analytic. Step 2. Appllcatlon of bifurcation theory: Thus applying the theorem (4.1.5) to (4.3.63) we find that the solutions of (4.3.59) near w = 0 coincide

192


with the solutions of the finite-dimensional system of equations (4.3.64) PB(w0 + g(wo), 00 + g(00)) = 0, where P is the projection of W, onto Ker L, wo is an arbitrary element of Ker L, and g is a complex analytic mapping of Ker L-+[Ker LIL in W,. Clearly, Ker L coincides with the harmonic (0, 1) forms so that P = H , and consequently the equation (4.3.64) is automatically satisfed since P B E HQ = QH = 0. Consequently, (4.3.60) shows that (4.3.52) will hold precisely if: (4.3.65a) (4.3.65b)

H [ w o + g(oo),wo + g(wo)] = 0, Qa[wo

+ g(oo), wo + g(wo)] = 0.

Step 3. Recapltulatlon: Finally, gathering our results together by showing that (4.3.65a) implies (4.3.65b) provided wo is sufficiently small, (so that the equations (4.3.65a) are the only obstructions to solving (4.3.59) near w = 0 with w = wo g(oo)).To prove this fact, suppose w satisfies (4.3.59) and (4.3.65a), then by (4.3.53)

+

Q Z [ w , w ] = 2Q[Sw, w ] = 2Q[ @[u,

01, a ]

Thus setting u = Q Z [ w , w] , we find u = -2Q[u, w ] . Therefore, by virtue of (4.3.63) and the fact that, for some absolute constant F > 0, 1141H

~ll~llHII~IIH*

Hence, for IJwIIH sufficiently small, llullH = 0, so that for w sufficiently small, (4.3.59) and (4.3.64) imply (4.3.65b). (This completes the proof of (*) of step 1.) Thus we find that if H 2 ( % , 0)= 0, the projection mapping H = 0 (in (4.3.65a)), and so the equation (4.3.52) has a family of solutions near w = 0 for each wo E HI(%, 0).More generally, if H 2 ( G x , 0)# 0, the equation (4.3.52) is solvable for fixed wo if and only if the system (4.3.65a) holds. This latter system can be interpreted as an “analytic set” depending on m , = dim H ‘(a, 0)complex parameters since fixing a basis wi for Ker L and setting wo = tiui, we find (4.3.65a) depends analytically on the complex variable t = ( t , , . . . , t,,). Moreover, it can be shown that the deformations Gx, just shown to exist are not complex analytically homeomorphic and “locally complete” in the sense that any other deformation near EXois equivalent to some %, constructed as above. For the details of this argument, we refer the reader once more to the monograph of Kodaira and Morrow (I 97 1).

4.4 ASYMPTOTIC EXPANSIONS A N D S I N G U L A RPERTURBATIONS

193

4.4 Asymptotic Expansions and Singular Perturbations

4.4A

Heuristics

'

Let i ( x ) be a C mapping of a Banach space X into a Banach space Y depending continuously on a small real parameter z. The following abstract situation is commonly encountered in physical problems associated with the solutions of the operator equationf,(x) = 0. There is a sequence x, (c) E X ( n = 0, 1, 2, . . . , N ) such that: (i) IIfL(x,(e))ll = O(z"+') for n fixed, as E-0; (ii) for small nonzero z, there is a solution X(c) of i ( x ) = 0 such that IlX(c) - x,(c)ll = O ( c n + ' )for n fixed, as E + O . Under these circumstances we say that x,,(c) is an asymptotic approximation to the solution X(c). Clearly, for E sufficiently small and fixed n, an asymptotic x n ( c ) provides as accurate an approximation to the solution X(c) as desired, even though in many important applications the sequence {IIx,(z)ll} diverges as n + co, for fixed E # 0 (cf. the discussion of (1.2.12) in Section 1.2B). Clearly, one must determine circumstances ensuring property (ii) of asymptotic approximation of a given sequence { x,(z)} satisfying (i) since numerical schemes for satisfying this latter property are well known. As a concrete problem in this connection, consider the semilinear Dirichlet problem

c2Au + u - g2(x)u3= 0, ~la= n 0, defined on a domain 8 in RN,where g(x) is a smooth strictly positive function of 8. We wish to investigate the solutions of this system for small z. In particular, we wish to justify the heuristic idea that the one-signed solutions obtained by setting z = 0 in (II,) (namely, uo(x) = ? g ( x ) ) should be "zero-order'' approximations to solutions of (II,) for small z if properly modified near a8 so as to satisfy the Dirichlet boundary condition on af2 Such a justification is relatively easy if N = I , g(x) = 1, and = (0, 1) since in that case an explicit solution in terms of Jacobian elliptic function' sn(x, k) is

(W

where l / r

= 2(1

+ k2)K(k) > n. Now, since

tanh

6

G

sn(& k) G 1

for 0 G

5G

K(k).

' Here the Jacobian elliptic function sn(5 - &,, k) is the solution of the differentia1 equation

I$ = (1 - u2)(1

- k2u2). The function sn(& k) is a periodic function of 6 with quarter-period K(k), with the same symmetry as sin t, and maximum value 1 at 5 = K ( k ) . Furthermore, sn(6, 0) = sin 6, K ( 0 ) = n/2, and as k f l with 6 fixed,

sn(& k)-tanh

(,

K(k)-ln{4(1

and K ( k ) is an increasing function on [0, I).

- k2)-"'},

194


Set &(c) = (1 - k2)'l2. As L J 0, k tad

((2 -

r 1, 6 - 4 exp(

- 1 / 2 3 / 2 ~then )

u,

&y2

I, the explicit justification analogous to that just described is impossible and so a quaiitative discussion is required. To this end we prove a general result in the next subsection and apply it to (II,) thereafter.

-

4.48 The validity of formal asymptotic expansions

The above example is clearly quite distinct from those described in the bifurcation theory earlier in the chapter. It is an example of a class of singular perturbation problems that can be posed abstractly as follows: Let f , ( x ) be a C ' mapping of the type just described. Suppose there is a sequence {x,,(e)} (n = 0, 1, 2, . . . , N ) of elements X such that (i) Ilf,(xn(~))II= O ( e " + ' ) for fixed n as e + O , but (ii) the linear operator f:(xn(0)) is not necessarily invertible for any n. Under what circumstances is x,,(E)an asymptotic approximation to a solution x ( e ) of f , ( x ) = 0 (in the sense defined in Section 4.4A)? Generally speaking, the asymptotic approximations x,(e) that are commonly used can be written as a power series in e, x,,(e) = ~ C : = o a iwhere ~ ' , a; may depend on e but llaill is bounded independently of e. Such truncated power series are called asymptotic expansions. In this case we shall assume hypotheses (IHIII) below in order to establish an answer to the singular perturbation problem just raised.

(I) There is a constant M independent of

e

and x , y , p E X such

that: (a) Ilf,<x +u) - 6 < 4-f:(x)yll G R ; and

Q

MllYl12 for all X,Y with IIxII, llyll

(b) Il(f:<x> -f:(r)>PII G Mllx -YII llpll for all llpll G R * (Note that this condition is automatically satisfied if the mapf,(x) has a uniformly bounded second derivative.) (11) There is an element xn(c) E X for all integers n Q N (a given e ' Ilajll G A; < 00, where A ; (but not integer) such that x,,(e) = ~ ~ - o a jwith necessarily a;) is independent of e and Ilfr(xn(e))ll = O ( ~ ~ ~ " + ' ) . (111) There are constants c, p > 0 independent of e and p such that for E > 0, f: ( x i ) is invertible and Ilf:(x;)pII

cepllpll

for some i

> p.

4.4 ASYMPTOTIC EXPANSIONSA N D SINGULAR PERTURBATIONS

(Notational remark: For convenience, the symbol appropriate norms in either X or Y.) We now state:

195

11 11 is used to denote *

Theorem Suppose f c ( x ) is a one-parameter family of continuously differentiable maps of X into Y for small nonnegative e, satisfying hypotheses (1)-(111). Then for N 2 2p and n < N -p, there is a p,(e) E X of f , ( x ) = 0 with llp,,(e)ll = O(en+') for solution Ec= x,(e) each n and each e in some interval (0, eo), where e,, is some small positive number. This solution X, is independent of n, and is the unique solution of f L ( x )= 0 such that IIx - xpll = O(ef'+'). (4.4.1)

+

Idea of the proof: The result is demonstrated in four stages:

+

(i) the solution pn of the equation f , ( x n p,) = 0 is rewritten in the form pn = Tfpn where T, is a bounded mapping of X into itself; (ii) for any integer n > N - p (in particular for n = N ) an application of the contraction mapping theorem to the equation p = T,p yields a unique solution pn with llpnll = O(c"-P+'); (iii) for any integer n < N - p , it is shown that (ii) implies that pn = pN + ZY-.+,aid satisfies p = T,p with llp,ll = O(c"+'), as required; (iv) for any two solutions p,, and p; of p = T,p both of order O(t."+') with n > p , their difference 6. = pn - p i satisfies the equation 6 = T,(p, + 6 ) - T,p,. This fact is shown to imply that 6, = 0.

+

Note that once (i)-(iii) are established Xc = x,,(r) pn is independent of n for 0 < n < Xf = xn(c) + pn and Xi = xm(e) + pm for 0 G m , n < N - p , Xe = Xi follows immediately from the definition of pn and p,,, given in (iii).

N - p since if

Proof: (i): theorem

First we show that by virtue of the hypotheses of the

Hence K(xJ is an invertible linear map for n 2 i : (4.4.3)

\ljy(x,,)Il

< (2/c)e-f'

Now let a tentative solution be denoted jT, = x,, + p,, and f , ( x , , ) = g,,(x, e), where 11 g,,ll < Z,, a constant independent of e. Hence we wish to determine p,, such that f,(xn + p,) = 0; i.e., we wish to solve (4.4.4)

L ( x n + p n ) - f c ( x n ) = en+'gn.

196


Now by virtue of hypothesis (I), the equation (4.4.4) can be rewritten (4.4.5)

f:(Xn)Pn

+ R,(x,,

P,) = En+'gn,

where (4.4.6)

MllPfllI2*

IlR,(xf19 P J I I

By virtue of (4.4.3), (4.4.5) may be rewritten (4.4.7)

P, =f:-'(~,){Efl+'gfl - R,(x,, P A . To obtain a solution p, of (4.4.7) we apply the contraction map(ii): ping theorem (3.1.1) to the map

T,P = f c - ' ( x , ) { ~ " + ' g ,- R,(x,, P ) )

acting on the sphere S ( 6 , s) = { p I p E X , llpll Q a€'}, where the numbers 6 and s are to be determined so that T, maps S ( 6 , s) into itself. To this end, we note the following estimates for p E S ( 6 , s):

II z '+ 'f:- I ( x,)g, II = 0 (E '+ llf:-l(xfl)R,(x,,

(by (4.4.3)),

-P)

(by (4.4.6)).

P)II = 0 ( E 2 " - P )

+

Thus if we choose s such that (a) n 1 - p 2 s and (b) 2s - p > s, T, will map S ( 6 , s) into itself for 6 sufficiently large (in fact 6 > 2Z,,/c) and z sufficiently small. Together (a) and (b) imply that s may be chosen so that n 1 - p > s > p . Thus we set s = n - p + 1 for any n > 2p. (Note that such an n exists since N > 2p.) On the other hand, for any p, p' E S(6, n p + 1)

+

IITCP - TCP'll

Q

ll.c-'(%IlllR,(xfl9

P ) - R,(x,, P".

Since (4.43)

R,(xfl, P ) - R, (x,, P') = d ( x , =/I[axfl 0

+ P)

-d(x,

+ P')

+ t P + (1 - 9 P ' )

-

f:(xfl)b- P')

- fc(X,)l[P

- P'l

dt,

applying the mean value theorem and hypothesis (i), we obtain IITCP - TCP'II

Q

I l f : - ' ( ~ , > l l ~ ~ ( ~ o l l P+ l l (1 - t0)llP'II)IIP

- P'II)

for some to E (0, 1). Thus since llpll, llp'll Q 8zfl-P+' and by virtue of (4.4.3),

11 T,p - T,p'II

Q ( iC E ~ ) - ~ ( M ~ E " - ~ +$11 ' ) I I ~ Q

2 M&"-zP+' C

IIp - p'll

with n

> 2p.

Thus by choosing zo < c / 2 M 6 , T, is a contraction map of S ( 6 , n - p + 1) into itself and so has a unique fixed point p, with IIp,II = O ( z " - P + ' ) . In particular for n = N we have (4.4.9)

llpNll = O ( P P + ' ) .

4.4 ASYMPTOTIC EXPANSIONS A N D SINGULAR PERTURBATIONS

(iii): Now suppose n < N - p , we find a p, such that f ( x n and llpnll = O(cfl+').Indeed let

+ p,)

197 =0

N

Pn

x

=

u,ci + PNI

i=n+l

then N

x

+ P,) =.f( x, +

f(Xn

%c'

+ PN

=.f(xN

+ P N ) = 0.

r=n+l

Furthermore, I=n+l

i=n+l

Hence JJp,,))= O ( c n + I )+ O ( c N - p + ' )and provided n (4.4.10)

< N -p,

IJpn/l= O ( c " + ' ) .

Thus for n < N - p , we have found the desired solution p,, to the equation (4.4.4) and furthermore the desired estimate (4.4.10). (iv): Finally, we demonstrate the uniqueness of p,, for n > p with

For n 2 2p, this fact is an immediate consequence of the fact that T, is a contraction mapping. Furthermore, for n 2 p , suppose there are two solutions p, and p, + 6, of (4.4.5) satisfying the estimate (4.4.10). Then from (4.4.7) and (4.4.8), p,, is a solution of the equation (Ip,I( = O(t:"+').

(4.4.11)

u = -f'-' e ( x n ) / l r ~ ( x n+ Pn +

- ~ ( x n ) l U

0

dt.

The uniqueness of p,, is a consequence of the following (4.4.12)

llull

Lemma Equation (4.4.1 1) has the unique solution u = 0 for sufficiently small.

= ~ ( c p + ' )for c

Proof: Set J ( u ) = J ; [ f : ( x , + p, + tu) - f:(x,)]u dr. Suppose u and u + S are solutions of (4.4.1 I), then K(x,)S = J ( u ) - J ( u S), so that by (111)

+

~ \ J (+ u 8) - J ( ~ ) l = l I~K(x,)SII2 + c E P I I S ( I . On the other hand, after suitable rearrangement, setting JI, (4.4.13)

= x,

+ p,,

I

J(0

+ 6) - J ( u ) =

J{ 0

+ r ( u + 6)) -K(.Yn + ru)lu

+ [Kf;vn+ r ( u + 8)) - KlS> dr.

So by hypothesis (i) IV(U

+ 6 ) - J(u)ll

MI1611 1 1 ~ 1 1+ M{ll4l +

IlSll + I l P n l l ~ l l ~ l l .

Assuming llSll # 0, and combining (4.4.13) and the above, we obtain t.cP

MII41 + M { l l c l l

+ 1 1 ~ 1 1+ IIP,II}

with

llP,ll

= O(cn+')5

198


which contradicts the fact that IIuII lemma is proved.

=

llSll = O(cP+').Hence 6 = 0 and the

Remarks: In case X = Y is a Hilbert space, the following condition can be substituted for hypothesis (111): (111')

(4.4.14)

there is a Banach space X' 2 X such that

> clllpllk9 (b) ( f ( x i ) p , P ) a c2cpIl~l12X- K l l ~ l l L

(a) ( f ( x i ) p , P)

where c l , c2, and K are positive constants independent of

E.

Indeed multiplying (a) by K , (b) by el, and adding, one obtains (CI

+ K)(f 0, for t + M and r = 0, 1, . . . , N 2. For both oalues of k there exist functions x,(t, c ) that appear to be approximate solutions of the problem. In fact, an obvious iterative scheme starts with xo(t) = g ( t ) and yields (4.4.16)

r2k

-

+

(4.4.17)

~2,,,(t,C )

= g ( t ) - c2kg"(r)

+ . . . + czm( - k)"g'*"'(t),

(0 < 2m = N ) , and an elementary calculation shows that

(4.4.18)

d2x2, ?k -+ x2,,,= g ( t ) + o dt2 x2,,,+0

(E~~ uniformly + ~ )in t,

as t - k o o .

However, for k = 1 this result is misleading. The general solution of the differential equation can be written explicitly, and on this basis it is straightforward to show that: (a) for k = 1, e # 0 and g = e-I2, say, the problem (4.4.16) has no solution; (b) for k = - 1, the problem has a solution X ( t , C ) and x~~ is an asymptotic approximation to

x.

4.4 ASYMPTOTICEXPANSIONS A N D S I N G U L A R PERTURBATIONS

199

We shall now show that both these conclusions are in accord with the (sufficient) conditions of Theorem (4.4.1). Choose X and Y to be the Sobolev space W , ,2 ( - 00, m), which is a real Hilbert space with inner product

Then ft(x) and j,(x) (which will be written f: in the present case, because here it is independent of x ) are implicitly defined by

1( 1(

( f , ( x ) , ‘p) =

-m

( x p , ‘p) = -cc

dx dV

- c2k

; I ; ;T; + xcp - g’p} di

- c2k

df di

dP d’p

+ p’p)

di

for all ‘p E X,

for all ‘p E X,

where we have used the Riesz representation theorem for linear functionals in Hilbert space and the fact that, for fixed x and p , the integrals are bounded linear functionals defined for all ‘p E X. It follows at once that condition (I) in satisfied, with M = 0. Condition (11) is also satisfied ) ~: the differentiability of the xlm, the Schwarz inequality, (we can define x ~ = x~ ~ +~Using and (4.4.18) in which the 0-term is actually c2rn+2( - k)rng(2rn+2)(f), we obtain

Therefore it remains to show that condition (111) is violated for k = 1 and holds for k = To establish the former, consider the function

-

1.

(4.4.20)

z ( t , c, p ) = {( p ) cos(t/c), where p is a small positive number and ((s) is a mollifier as follows: { E Cm(- m, m), {(s) = 1 for Is1 6 I , {(s) = 0 for (sI > 2, and 0 < ((3) G 1 for all s. Clearly, z E X, and if we can show that

(4.4.21)

t t L z t-0t 11z11

as p + O

(with c

fixed, and k = l),

then no inequality of the form (111) is possible. To prove (4.4.21), we have (cf. the derivation of (4.4.19))

upon substitution of (4.4.20). Also

=J-l:;p(

(f

sin

:)2

and so (4.4.21) is proved. To show that (111) holds for k = -1, X’= L2(- m, 00); for all p E X we have (f’p, P )

IPI:.,

and

+ cos2

-

di

>2

for

e

< 1,

we refer to the alternative form (1113, with

( f : ~ ,P )

~211~ll;- IIPII?~,.

200


4.4C Application to the semilinear Dirichlet problem (I&)

In order to illustrate the applicability of Theorem (4.4. l), we consider the semilinear Dirichlet problem (II,)mentioned earlier. The following fact can be established: (A) For sufficiently small values of c, the problem (II,) has a unique smooth positive solution u ( x , c) that tends to l/g(x) as c + O , outside a narrow “boundary layer” of width O(c) concentrated near an.

This result is proven by (B) constructing an approximate solution U M ( x ,e) that u ( x , e) - U,,,(X,e) is O ( c M + ’ uniformly ) on (for any integer M ) . Then we show that for M = 0 this expansion is indeed asymptotic to a true solution of (n,) for e sufficiently small, and consider the behavior of Uo(x, c) for small e. In order to apply (4.4.1), we construct approximations Um(x,c) for m > 1 satisfying

a

= ~ ~ = O c m ~ m e) (such ~ ,

(4.4.22) K,( U,,,) = c2AUm + U, - g ’ ( x ) U i uMl?JS2

=

O ( c M + ’ ) uniformly in D,

= O’

To display the principal ideas in the construction of such approximations, we describe the steps leading to the lowest approximation Uo(x, c). (a) By neglecting c2 AM in the differential equation K,u = 0, we find the approximation v o ( x ) = l/g(x), which is expected to differ from the positive solution u by O(e’) for cJ,O with x bounded away from aD. (b) To describe u near the boundary aQ, we first attach labels ( s , t ) to the points in a fixed neighborhood D. = { x I 0 < t < t . } of a D ; here s = xo E a D labels the normal to a D through the point xo, while t measures distance from aD, as in Fig. 4.6. For explicit calculations, s is replaced by an (N - 1)-dimensional surface coordinate u. Making the stretching transformation t = €7,we then seek a function wo(s,r ) which is to approximate u in the “boundary layer”; more precisely, u - wo is to be O(c) for cJ,O with 7 fixed (so that tJ,O). Writing the operator K, in terms of s, 7, and c and keeping only the dominant terms of this form of K,, we obtain the boundary layer problem (4.4.23a)

a h0

- + wo - gl(s, 0)w; = 0, a72

where g.(s, t ) = g(x), and where the condition for rtco is a “matching” condition, suggested by the fact uo(x)+ l/g.(s, 0) as t i 0 . The solution of

4.4 ASYMPTOTIC EXPANSIONS A N D SINGULAR PERTURBATIONS

201

n

FIG. 4.6 Illustration of the boundary layer phenomena for (rIc).

the problem (4.4.23) in which s has the role of parameter is

(c) Because lim,Louo(x) = 1irnTTm wo(s, T) (a particular case of what is often called the “asymptotic matching principle”), we can now use the prescription for forming the leading term of the “composite expansion.” We define (4.4.24)

U;(x, E )

= uo(x)

=- 1 dx)

+ wo(s, f

( ’+

) lii

g4s7 0 )

-

wo

t tanh €21’2 - 1)

on a*: 0 < t < t.. To overcome the trivial difficulty that t is not uniquely defined on Q - Q., we note that tanh(t/e2’l2) - 1 is transcendentally small for cJ0 with t fixed and positive, and introduce a mollifier {(x) E C“(Q) such that {(x) = 1 on 0 < t < 1/2t., {(x) = 0 on Q - Q,, and 0 < 5 < 1 on 32. We can then define 1 t (4.4.25) U ~ ( XE ) , = - + (tanh - - l), d.1 g*(s, 0 ) €2’12 and it is readily verified that this satisfies (4.4.22) with M = 0. This is because of the dual role of the final term in (4.4.24) which to the lowest

202

4


order cancels uo in the boundary layer, cancels wo at points bounded away from a!2, and makes Uo more accurate than either uo or wo in the intermediate region defined by t = a(€), where 0 is any function such that t = @(€)LO and T = c-'e(e)Tco as €10. For higher approximations U,, M 2 1, the constituent functions in (4.4.24) are replaced by finite series M

2 m=O

M

€"um(x),

M

eewe(s, T), p=O

and

P

2 ce 2 p=o

W~,,,(S)T",

n=O

respectively. The proof of (4.4.22) is then complicated by the fact that the coefficient functions urn,we, and we,,, are defined by elaborate recurrence relations. Now assuming the approximations U M ( x ,c) have been constructed, we must reformulate the Dirichlet problem as an operator equation f ( x , e) = 0, in appropriately defined Banach spaces X and Y , in such a way that we can verify the hypotheses (I)-(111) of Theorem (4.4.1). The duality procedure described in (Section 2.2D) shows that (at least for N < 3) the solutions of (IT,) can be regarded as generalized solutions of an operator equation of the form L,u + Nu = 0, where L, and N are bounded mappings of implicitly by the formulas:

into itself defined

Now if one defines f , ( u ) = Leu + Nu, then hypotheses (I) and (11) are easily verified since the second derivative of f , ( x ) is uniformly bounded in any sphere ( ( U ~ I ~ ~%’J(vp)*, az

is independent of c and p.

Prool: The primary steps in the proof are, first, a sharp form for the Poincare inequality, and secondly, a simple approximation to the function U,. (i): Writing p as a line integral along the inward normal to afl, and using the Schwarz inequality, we have

Integrating over the subset of

a. determined by 0 < t < I, we obtain

(4.4.29) where H ( 1 ) is a continuous function resulting from the curvature(s) of as 1-0. (ii): We define

h,(x,

C) = 3 -

an, such that H ( I ) +

1

3g2u;.

Then one shows that (in terms of the mollifier { ( x ) defined before (4.4.25))

1 h , - { ( x ) sech2 3

$

=

{(I - {)( 1 - tanh

$ )’+

O(C),

and at those points (f./2 < t < f.) where {(I - 3) # 0, the function 1 - t a n h ( ~ / 2 ’ / ~ is ) exponentially small for cJO. Accordingly (4.4.30)

h,(x,

C)

= 3{(x) sech’

$ + r,,

where k = k ( M ) is a constant independent of x and (iii): We can now estimate the functional (eM,.P,

P)

=j-&fz(vP)2

ac refers to the complement of uniquely defined on $2 - $2.). Then

f

< uc in the whole domain fl (even though f

is not

204

4


Now (2/a2H(0)) - 1 > 0 for a < 2I/*, and 2 - 3 s e ~ h ~ ( a / 2> ~0 / ~for ) a > 321/2. Choose a = 2 2’1’ = a,, and c, so small that, for 0 < c < co,

where p ( M ) is independent of c. Then (4.4.31)

(e,,,p,

p)

for 0 < c < cO.

> p2Lp2

(iv): Since (4.4.30) implies that 2 - h,,, >

for 0 < c < c,. Multiplying this by p2/(1 kc, + p2), we obtain the result (4.4.27).

-1 -

+ kc,),

kc, we also have

adding (4.4.31), and defining v 2 = p2/(1

+

Thus the hypotheses of Theorem (4.4.1) are completely verified so that since the approximations U,,,(x, E ) can be computed for M y 2, Uo(x,E ) is an asymptotic approximation to a true solution in the W , , , ( Q )norm. However, there remains the problem of verifying (A) in the pointwise sense. The statement (A) follows immediately from the form of Uo(x,E ) discussed in (4.4.25), provided we show that Vo(x,E ) is an asymptotic approximation in the C(n) norm. For N = 1, this fact follows from < const. 11 Uo(x, ,, , Sobolev’s inequality (1.4.1); indeed, 11 U o ( x , where the constant is independent of E . For N = 2, 3, the result follows the details of the proof of (4.4.1) and the L, regularity theorem for linear elliptic equations of Section 1.5. Indeed, utilizing the notation and results of Theorem (4.4.1), (4.4.32)

u ( x , E ) = Uo

+

3

q . e i + p4,

i= 1

where IIp4111,2= O(c3).The L, regularity theory implies that for N = 3, p4 is a generalized solution of the linear equation e 2 Au = f(p4) withf E L,(Q) and Il fllo, 2 = 0 ( E 3 > . Consequently, llP4112,2 = E -211f(P4)110. 2 = O(E). Now the Sobolev imbedding theorem yields the estimate llp411c c ~=) O ( E ) so , that (4.4.32) implies l u ( x , E ) - V0l = O ( E ) . Remark For N > 3 and nonlinearities that grow faster than u3, an exact analogue of the result (A) holds. See the Notes at the end of the chapter for references and the idea of the proof.

4.5 Some Singular Perturbation Problems of Classical Mathematical Physics

Many problems P, in mathematical physics can be made to depend smoothly on a small real parameter E , in such a way that when E = 0 a

4.5 SOME SINGULAR PERTURBATION PROBLEMS

205

solution xo to Po can be exhibited explicitly. One then attempts to show that the small simplibing change in passing from P, to Po has a correspondingly small effect in the solution X ( E ) of P,; i.e., the problem P, has a solution X ( E ) = x o + o ( l ) as E -0. What actually occurs in a broad class of important problems P, defined by equations of the typef,(x) = 0 is that a formal solution X ( E ) =CpO=oal~l forf,(x) = 0 exists with x(0) = a. satisfying f o ( x ) = 0. However, such series often actually diverge for E # 0 since the magnitude of the coefficients a,, do not tend to zero. Nonetheless, one hopes that if X ( E ) is truncated to x,,(E)= x ~ = O a lthen ~ ' rx , ( E ) is asymptotic (in the sense mentioned in Section 4.4A) to a true solution Z ( E ) of k ( x ) = 0. In this section we point out three problems of this class, in which this asymptotic property of a formal solution can be proven, on the basis of Theorem (4.4.1). In all the cases considered below, we note that (4.4.1) reduces the problem to finding sharp bounds for the norms of certain linear operators. 4.5A

Perturbation of an anharmonic oscillator by transient forces

Consider the ordinary differential equation (4.5.1) i + x = f ( x ) + Eg(t); f ( x ) = 0 ( ( x l 2 ) at x = 0 , where g ( t ) is a continuous function on ( - 00, co) decaying exponentially at 00, and E is a small parameter. We are interested in the behavior of the solution x ( t ) of the Cauchy problem with zero initial conditions for (4.5.1) as t -+ 00. This solution does not tend to zero as t -+ 00, at least for small \el. Actually what can be proved is the following. (4.5.2) For E sufficiently small and f ( x ) real analytic near x = 0, the solution x ( t ) tends asymptotically as t + cc to a periodic solution u ( t ) (near x = 0 ) of the equation i+ x = f(x). Instead of giving a proof of this result, it is perhaps somewhat more appropriate to discuss it in the context of Section 4.4A. If, using a majorant method, one attempts to compute the solution x ( t ) of (4.5.1) as a power series in E , x ( t ) =x,,"=,a,,(t)~", one finds that the coefficients a,(t) are unbounded functions of t over the interval [0, 001. A different method that does succeed consists in denoting a possible solution x ( t ) of (4.5.1) as x ( t ) = u ( t , A, 6 ) + y ( t ) , where u ( t , A, 6 ) is the solution of (4.5.1) with period X and phase b, and such that y ( t )-+ 0 as t -+ cc. Next one finds formal asymptotic expansions for the period A(€), the phase b ( ~ )and , the remainder y ( t ) . One then justifies these asymptotic expansions on the basis of (4.4.1). In fact this is the approach used in a paper by Ter-Krikorov (1969), where the interested reader will find detailed proofs.

206

4


4.58 The membrane approximation in nonlinear elasticity The partial differential equations governing the equilibrium states of a thin elastic plate naturally contain a small parameter c2, a measure of the plate’s thickness (cf. (1.1.12)). The membrane approximation to the problem of determining the equilibrium states under given body forces consists in setting c2 = 0 in the above-mentioned equations, and finding the solutions of this reduced system. More explicitly, let 3 be a bounded domain in R 2 with boundary a 3, then the partial differential equations defined over 3 can be written in the form

(4.5.3)

A2F + 4 [ w , w ] = 0,

c2 A% - [ w , F ] = g ,

where the bilinear form [f, g ] = fxxgyy+ f,g,, - 2fxygxy. The physical quantities F, w , and g have been described earlier. For simplicity, we assume that the plate is clamped, so that a given problem is specified by adding the boundary conditions

(4.5.4)

D%laa

=0

for

0 in 3. It is easy to show that such positive solutions are unique (if they exist), and in fact using the techniques of Part I11 one can show

207


that they do exist in a large class of elastic problems. More importantly, the following result is known. (4.5.6) Theorem Suppose a positive solution (w,, F,) for the system (4.5.3) - (4.5.5) exists, then for e > 0 sufficiently small the system (4.5.3)(4.5.4) has a unique positive solution ( w e , F,) such that (we, F,)+(wo, F,) uniformly apart from a narrow region of !J near a i l . The proof of (4.5.6) is parallel to that of (A) of Section 4.4C. One first constructs formal asymptotic expansions for functions w,(x, e) and F,,,(x, e) so that they satisfy the boundary conditions (4.5.4) exactly and . one represents the the equations apart from terms of order O ( e m + ’ ) Then solutions of (4.5.3)44.5.4) as solutions x = ( w , F ) of an operator equation of the form d ( x ) = 0 acting on the Sobolev space W2,*(a) exactly as in (2.5.7). The Sobolev inequalities show that the second derivative f c x x ( x ) is uniformly bounded on sets llxll = ( I I W I / ; , ~ + ~ ~ F ~ ~ ~ c/,l V w12. Furthermore, by the construction of the approximations Fk, sup, I D“(Fk - F,)I = O(le() for la1 = 2, and k 2 0, so that ( f : ( x k ) y , y ) > fe21/y11;,2for e sufficiently small. Therefore, as in the argument following (4.4.14), by the Lax-Milgram theorem f : ( x k ) is invertible and Ilf:(xk)ll > t e ’ . Thus the hypotheses of (4.4.1) are verified and Theorem (4.5.6) is proven, once one notices that the leading terms in the asymptotic expansions for Fk and wA are (w,, F,). For more details, see Srubshchik (1964). 4.5C Perturbed Jeffrey-Hamel flows of a viscous fluid The Navier-Stokes equations defining the steady plane radial flow of a viscous incompressible fluid between two inclined planes (meeting at an angle 2a) admits exact solutions. The solutions, known as Jeffrey-Hamel

208

4


flows G ( a , R ) , exist for all Reynolds numbers R , and for given parameters (a and R ) are nonunique (in general). Most of the Jeffrey-Hamel flows

exhibit a combination of inflow and outflow along a profile. Thus it is of interest to consider a perturbation of the geometric situation defining these flows with a view to proving that among the special Jeffrey-Hamel flows there is one that is a first approximation to a solution of the perturbed problem. As was observed by L. E. Fraenkel, an interesting perturbed problem consists of a class of two-dimensional symmetric channels C with walls whose radii of curvature are uniformly large relative to the local channel width. Fraenkel proved the existence of a unique Jeffrey-Hamel flow G,(a, R ) depending analytically on a near a = 0, and he constructed of the Navier-Stokes equations a formal solution' u ( x , t) = Go EC,"=,tn#,, in C in terms of the small curvature E of the walls of the channel. For appropriate values of the physical parameters R and a this formal solution exhibits a separation phenomenon in which the velocity field of the flow is divided into distinct regions of forward and reversed flow by a zero velocity curve, which is itself separated from the wall of the channel. An important mathematical problem in this connection, (solvable by using the methods of (4.4.1)), is the justification of the use of the formal solution u ( x , t) as an approximation to a true solution of the physical problem. The formal solution u ( x , E ) mentioned above cannot be expected to converge for c # 0 since repeated differentiation of certain functions in the construction of u ( x , E ) makes the coefficients of t n roughly comparable with n!. Thus it is natural in this situation to show that the truncated formal solution is an asymptotic approximation to a true solution of the Navier-Stokes equations in the sense of Section 4.4A. To formulate this problem mathematically, we proceed as follows. Suppose the channel C is the image of the strip S2 = R ' X ( - 1, 1) under the conformal mapping z = z(w, E ) characterized by the equation d z / d w = he" and the function a ( c w ) = (d/dw)(log(dz/dw)). Thus with z = x + 9 and w = u iu, Jdzl = hldwl, and a ( t w ) turns out to be approximately the angle which the upper channel wall makes with the x axis. Then the "vorticity form" of the Navier-Stokes equations becomes

+

+

(4.5.7)

[A+R(h

a -4" G a ] ]2 1

A$=o,

where # is proportional to the stream function and A is the Laplacian with respect to ( u , u). Setting h, = hk and h, = hh, and using the fact that

'

The parameter a is a fixed parameter in the Jeffrey-Hamel problem, but a slowly varying function in the perturbed problem.

209


A(1og h ) = 0, we find that (4.5.7) can be rewritten as (4.5.8)

L(#)

G

This equation is supplemented by the boundary conditions (4.5.9)

#=

(4.5.10)

qU+O

GU = 0

1,

as

for u = 2 1,

I u( +oo.

If a is a real constant, k = a , h = 0, and # is independent of u, then (4.5.8) becomes (4.5.11)

#uuuu

+ (4a2 + 2Ra#o)$uu= 0.

Solutions of the two-point boundary problem (4.5.9) and (4.5.11) are precisely the Jeffrey-Hamel flows mentioned above. To define the formal solutions u ( x , e ) we set u = U / E in (4.5.8) and suppose I#,I = O ( E )SO that k = a ( u ) + 0 ( c 2 ) , while X = O(t-). Then (4.5.8) can be written as #uovu

+ 4ffZ4Juo + 2Ra$o$oi,,= O (e) .

Now if we assume that #(x, E ) = G,(u, R, .(a)) + C,"=,E"#~, then the #,, can be computed iteratively, each being the unique odd solution of L# = 1csuuu + 4ffvoo+ ,,a(

1)=#,=0

aG0

).

= Ffl(G0, #,> *

'

. #,,-A 3

U

for u = + I ,

where F, is found from the exact equation (4.5.8). Now we note (without l ~ n # n constructed proof) that the truncated formal solution Ir/, = Go as above is an approximate solution of (4.5.8) in the sense that in the notation of Section 4.4A:

+ zf=

(i) fr(qN) = O ( E I) uniformly on a; \ k , satisfies the boundary conditions (4.5.9), (4.5.10); (ii) Ilf,(SN)[[L,ca, Q kN(R, a ) ~ ~ + for ' / c~ sufficiently , small. (iii) +

We can now state (4.5.12) For E sufficiently small and the quantities R,a suitably restricted, (4.5.7) has a classical solution # = Go + po. Furthermore, $ is the

210


only solution such that 1) - Go(( = O ( E ) .Here Go = Go(a, R ) is the Jeffrey-Hamel flow mentioned at the beginning of Section 4.5C. Proot: We apply (4.4.1) with p = 0, n = 1, and N = 1 by rewriting (4.5.8) as an operator equation in the closed subspace H of odd functions belonging to W2,2(Q).An appropriate norm for H can be chosen to be lu112H-

x JnlW.

lal-2

Indeed by replacing 4 in (4.5.8) by Go + p, we may suppose p satisfies a null boundary condition. Clearly by arguing as in Section 2,2D, the equation (4.5.8) can be then rewritten as an operator equation in H of the form f , ( p ) = Lcp+ RN,p= U ( c ) , where L and N are bounded continuous mappings of H into itself defined formally by the equation L(G0 + P ) = f,(Go) + L,P + NJP)

so that the linear operator L, is implicitly defined by

A41 + W R ) .

( 4 4 , 'p) = i [ ( A 4 ) @ ~+) A4(4kcp, + 4ApJ + 4(k2 +

Clearly llN,+ - N,'pll < k(ll4ll + Il'pII)II$ - 'pII for some constant k independent of 4 and cp, and N satisfies hypothesis (I) of (4.4.1). Thus by (4.4.1), to prove (4.5.12) it suffices to prove that (a)

Le is invertible and IILc+\l > kll+ll, where k is a constant independent of

c.

By virtue of (1.3.21), ( 8 ) will be established once we find a positive constant k , independent of such that (L& 4) > k,11+1&. Now a simple computation with p = 4a + R ( a G o / a u ) and q = 4a2 2Ra(GO/3u)shows that c

+

(LA

4 ) = Jnt lA4I2 +

( H U

Integration by parts using the fact that

(t)

(Lc4,4) =&A412

+ &)A4

a = a(cu)

+ 4"\C.l; + P,4+u)

+ O(c)ll+IIZH.

gives

+ R(a2Go/ao2)+4"" - q(4: + 4 3 ) + O(c)ll4IlZH.

Thus it suffices to find a suitable lower bound for the first quadratic form on the right-hand side of (t).To this end, by suitably restricting R and a , we shall find a constant h = A(R, a ) > 0 independent of c such that for all odd (in u) functions 4 E Com(Q), (4.5.13)

1

Q k 4 )= J l(2rC:u + 4:u

+ RG,IW;, - q(4: + 4:)) do

> hJ-L1(2+:u + 4 3 do. Once this inequality is proved, (*) is established by adding q ,: to both sides of (4.5.13) and integrating with respect to u. To prove (4.5.13), we note that the inequality (4.5.14)

1-,(do- q'p3 do > PJI

1

$ du

for

'p E ~ 2 , 2 ( 1,

1)

can be established by suitably restricting R and a and if 'p is odd in u, p can be increased to p, (say). Applying this inequality to 4 and / Y do, we find that

/ ( 4 & - d;2) > PlJ4.'.

/M"- 94:) > pJ3:.


21 1

Then arguing as in (4.4.14), these two inequalities imply the existence of positive constants p 2 0,

It remains to choose 6 so that 1 + p2 - p6 > 0 and p3 - p / S > 0. Clearly this can be accomplished by restricting R and a so that / I 2 < ( I + p2)p3. Thus it remains to investigate the validity of (4.5.14) for some fixed > 0. To this end, it is necessary to investigate the lowest eigenvalue of the linear eigenvalue problem w(’”)

+ (qw’)’+ Aw” = 0,

w = w’ = 0,

u=

? 1.

A continuity argument shows that p > 0 for all R > 0 provided a lies in the interval where the formal approximation Go(a) IS the unique Jeffrey-Hamel flow depending analytically on a.

For details see Fraenkel (1973).

NOTES A Linear stablllty of bifurcated branches of solutions As was mentioned in Section 4.1 there is often an “exchange of stability” in physical

systems after a bifurcation. In the elasticity result of (4.3.36) this was demonstrated by energy considerations. In general for non-Hamiltonian systems the less precise linear stability criterion, mentioned in Section 4.1 is useful. This criterion is baszd on information concerning the spectrum of the linear operator f x ( 2 , i) at a solution (2, A) of f ( x , A) = 0. The LeraySchauder degree theory is often useful in this connection. Thus, for example, in finitedimensional system: linear stability is determined by proving that the real parts of the eigenvalues of fx(2, A) are negative, while instability results from any eigenvalue with positive real part. Thus if we compute the Brouwer degree of f ( x , A) at 0 as a function of A, as A crosses a bifurcation point we shall be able to ascertain facts about the spectrum of fx at any solution ( 2 , A) of f(2,A) = 0 since the Brouwer degree can be computed additively by linearization at nonsingular solutions. A similar results holds for the infinite-dimensional case. For further information the interested reader is referred to the paper by Sattinger (1971). B Blfurcation for general operator equations at elgenvalues of odd multlpilcity

Letf(x, A) be a C ’ mapping of a neighborhood of (0,0) of the Banach space X X R into Y such that (i)f(O, A ) = O for all A near 0, ( i i) f , ( x , A) is a C ’ function of I , (iii) the linear operatorf,(O, 0) is a linear Fredholm map of index zero and dim Kerf,(O, 0) is odd, and (ii). For x E Ker fJ0, 0), fxx(O, 0 ) x $Range i! f*(O, 0). Then, as a generalization of the results (4.1.12) and (4.2.3), one can prove that (0,0) is a point of bifurcation relative to the operator equation f ( x , A) = 0. This result is obtained by decomposing the Banach space as in (4.1.12) and applying the properties of the Brouwer degree to the associated bifurcation equations. The full details are given in Westreich (1973). C Reductlon of the blfurcatlon equations under symmetry assumptions

In many bifurcation problems of the multiplicity rn of an eigenvalue X of an associated linear problem is caused by invariance properties of the associated operator equation under a

212

4


group of isometries. In many cases this symmetry allows a reduction in both the number of equations in the bifurcation equations and in the number of unknowns in this equation. Thus in the secondary steady flows of the Navier-Stokes equation associated with convection in a horizontal fluid heated from below (the so-called Benard problem), the secondary solutions observed have a hexagonal cellular pattern. This result can be obtained from the bifurcation theory developed in Section 4.1 by finding solution of the associated nonlinear boundary value problem in a Banach space of vector-valued functions which themselves possess “hexogonal” symmetry. This was first carried out in Judovitch (1968). A general study of this situation was carried out by Loginov and Tregonin (1972). D Boundary layer phenomena for remlllnear Dlrlchlet problems A generalization of the result given in Section 4.4C can be carried out for boundary value problem defined on Q c R N

(t)

C ~ A+Uf ( x , U )= 0,

ulaa = 0,

wheref(x, u ) is a C ” function of its arguments with the properties (i) there is a C“ positive function T ( x ) defined on a such thatf(x, T ( x ) )= 0 on 3i and (ii)

f,(x, T ( x ) ) < 0 on

with

lT(l)(s,

y ) dy

>0

for fixed u E [0, T ( x ) ] .Theref(x, u) is not subjected to any growth restriction. Then, in order to justify the approximation uo = T ( x ) for a positive solution f ( x ) apart from a small boundary layer of width O(c) near aQ as c+O, one can work in Holder spaces C”*P(Q) using the procedure of Section 4.4B. However, in order to obtain the crucial estimate for the form of the linear operator L = [fs’(ui)]-’ the Sobolev space context is essential. Once an estimate for L has been obtained, the Sobolev inequalities give a pointwise estimate for L when regarded as mappings between the appropriate Holder spaces. For the complete details the interested reader is referred to De Villiers (1973). See also Fife (1973). E Blbllographlc notes

Section 4.1: Poincare’s original paper on bifurcation theory can be found in Poincare (1885). This paper was devoted to determination of equilibrium forms for a rotating ideal fluid was inspired by a number of conjectures in the treatise by Kelvin and Tait (1879). Later treatments of this problem include Liapunov (190&1914), Lichtenstein (1933), and Appell (1921). A comprehensive modern study is still noticeably absent. As mentioned in earlier notes, Liapunov’s criterion (4.1.4) is generally proved by the majorant method, see Siege1 and Moser (1971) for a modem treatment. The reduction of bifurcation theory to a finitedimensional problem as in (4.1.5) is generaly called the Liapunov-Schmidt method due to the fundamental papers Liapunov (1906) and Schmidt (1908). A recent book on this subject is Vainberg and Tregonin (1974). Our treatment of simple multiplicity (4.1.12) is due to Diustermatt, and is intended to suggest the importance of the techniques on the recent theory of singularities in more difficult bifurcation problems. Other recent treatments of bifurcation in the case of simple multiplicity include Crandall and Rabinowitz (1971) and Westreich (1972). Krasnoselski’s books (1964) yield very detailed information on this problem. The relationship between bifurcation theory and nonlinear normal modes can be found in Berger (1969). The iteration scheme of Section 4.1D can be found in Berger and Westreich (1974). A survey of constructive methods in the higher multiplicity case can be found in Sather (1973), and in numerous other papers and books. Unfortunately, such methods often are not useful in practice since definite results are obtained by making assumptions that are not easy to verify.

NOTES

213

Section 4.2: The use of transcendental methods in bifurcation theory is well described in Krasnoselski (1964), Berger (1970a) and Cronin (1964). In particular the use of the degree of a mapping is due to Krasnoselskii (1964) who also noted the importance of gradient operators to obtain sharp results on bifurcation points that are independent of multiplicity. Prodi (197 1) seems to be the first paper utilizing Morse theory in bifurcation problems. See also Berger (1973). The sharp results on bifurcation theory for complex analytic mappings such as (4.2.4) were described in Cronin (1953). Our proof follows the paper Schwartz (1963). The use of the Liusternik Schnirelmann theory of category in bifurcation theory is described in the papers Berger (1969, 1970). Recent interesting results on applying the higher homotopy groups of spheres to bifurcation problems involving more than one parameter can be found in Ize (1975). See Amer. Math. SOC.memoir # 174. Section 4.3: A good survey of the periodic solutions of the restricted three-body problem near the equilibrium points L,-L, can be found in the article of Deprit and Henrard (1969). The treatment of buckling phenomena in nonlinear elasticity given here follows Berger (1967). Our discussion of the onset of turbulence for viscous fluids as a bifurcation phenomena follows Judovitch (1966, 1967). Other useful results include Kirchgassner and Sorger (1969) and Gortler el ol. (1968). The Taylor vortices were first described in Taylor (1923), but their mathematical study via the full nonlinear Navier-Stokes equations began with the papers Velte (1966) and Judovitch (1966). Our discussion of the bifurcation of complex structures on higher dimensional complex manifolds follows Nirenberg (1964) and Kuranishi (1965). Recently Kuranishi has used the Nash-Moser implicit function theorem to obtain results on deformations with singularities. An alternative approach to this problem using nonlinear functional analysis can be found in a recent paper by Forster (1975). The purely algebraic approach to this problem suffers from the fact that the formal power series solutions constructed generally diverge. Secfion 4.4: The result (4.4D) is adapted from the papers by Berger and Fraenkel (1969, 1970). The method for more general elliptic boundary value problems has been further developed by Fife (1973). Section 4.5: An excellent survey of singular perturbation problems in mathematical physics is Friedrichs (1955). The result on the perturbation of periodic solutions of an anharmonic oscillator is due to Ter-Krikorov (1969). Our discussion of the validity of the membrane approximation in nonlinear elasticity is adapted from Schrubshik (1964), while our discussion of perturbed Jeffrey-Hamel flows can be found in papers by Fraenkel (1962, 1973).

F Normal Modes for Nonlinear Hamlltonlan Systems As described in Section 4.2, transcendental methods of bifurcation theory are crucial in establishing general results on the preservation of normal modes of a linear autonomous Hamiltonian system of ordinary differential equations under a nonlinear Hamiltonian perturbation of higher order. This situation is described for second-order systems in the papers of Berger (1969, 1970a, 1971~).Berger’s methods have been adapted to first-order systems in a recent unpublished paper of Westreich, although in this case certain restrictions appear due to an example of Siegel, see Siegel and Moser (1971) pp. 109-1 10. Interesting finite-dimensional approaches to the problem for first-order systems can be found in Weinstein (1974) and Moser (1976) who both use the Ljusternik-Schnirelmann transcendental method analogous to Berger (1970a). See Section 6.7. A possible extension to hyperbolic partial differential equations is described in Berger (1973).

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PART Ill

ANALYSIS IN THE LARGE

The aim of Part 111 is the extension of the local results described to a global context in such a way that the specific problems (discussed in Chapter 1) can be successfully treated. In order to accomplish this goal, methods mixing analysis with topology will prove quite useful. Such transcendental methods were quite valuable for the bifurcation theory of Chapter 4.For global problems, however, this mixing is often essential for a proper understanding of nonlinear phenomena. It is this combination of analytic and topological techniques that has lead to the highest achievement in our subject. In practical terms, the extension from local to global is particularly important for two major ieasons. First, a sufficiently accurate first approximation to the solution of a given problem may not be available. Secondly, although such an approximation may exist, many problems require the totality of solutions S of a given problem to be considered as a whole. Indeed the set S is often partitioned into distinct classes { S,} with various classes of prime importance under differing circumstances. Rather than put forward a very general global theory, we shall be content to describe a realm of ideas midway between classical linear functional analysis and the theory of nonlinear functional analysis on general infinite-dimensional manifolds. We now summarize (in the abstract): The problems to be discussed. To fix notation, let f be a bounded mapping defined on an open domain U of a Banach space X with range in another Banach space Y . Then we pose the following questions: (i) (Mapping problem) Under what circumstances is f surjective (i.e., f ( U ) = Y), univalent, or a homeomorphism of U onto f( U)? (ii) (Linearization problem) What global properties of f can be deduced from the local behavior o f f ? In particular, i f f E C ' ( U, Y ) , what global properties can be deduced from f'(x)? (iii) (Solwabiliy problem) Determine necessary and sufficient conditions for the solvability of the operator equation f(x) = y For fixed y E Y , analogous to the classic linear Fredholm theory. 215

216

PART 111 ANALYSIS IN T H E LARGE

(iv) (Problem concerning global structure of solutions) Can one obtain a description of the set of solutions of the operator equation f ( x ) = y for fixed y E Y? In particular, under what circumstances does the equation have at least a given number of solutions? (v) (Problem concerning classifcation of solutions) Determine a classification of solutions of the operator equation f ( x ) = y that is invariant under small (suitably restricted) perturbations of the mapping f. (vi) (Deformation problem) Under what circumstances can any of the above problems be answered by smoothly deforming the given mapping f E M( (I, Y) to a simpler mapping f i .E M (.U , Y ).? _ (vii) (Problem concerning parameter dependence) Suppose the given mapping f depends continuously on a parameter A, then exactly how do the solutions of !(A) = 0 depend on the parameter A? (viii) (Approximation problem) Under what circumstances can the properties of the mapping f acting between infinite-dimensional spaces be deduced from finite-dimensional approximations for the range o f f ? (ix) (Generalized operator theory) What parts of the theory of linear operators in classical functional analysis extend directly to a nonlinear context? (x) (Problem of nonlinear effects) What qualitative features of the operator f affect the answers to the above problems? (xi) (Problem of infinire dimensional effects) What answers to the above questions are related to the infinite dimensionality of the Banach spaces X and Y ?

Typical of the situations we consider are nonlinear eigenvalue problems of the form A x = hBx, where at least one of the operators A and B is nonlinear. The problem is to find the totality of nontrivial solutions (x, A) of this equation, and to classify these solutions in a manner consistent, if possible, with linear eigenvalue theory. Since the solutions we seek are not local in nature, the methods of Part I1 must be supplemented by more all-encompassing ones. The interesting nonlinear phenomena for these problems (e.g., “continuous spectra” (see Fig. 6.3) p. 381) thus provide motivation for the development of deeper methods of study.

CHAPTER 5

GLOBAL THEORIES FOR GENERAL NONLINEAR OPERATORS

The study of mapping properties of general nonlinear operators calls for new methods of study quite distinct from the theory of Part 11. Here three such general methods are discussed. The first, linearization, is based on piecing together local information about the Frechet derivative f ’ ( x ) of a mapping f E C ’ ( X , Y ) by considering the geometry of the image f(X). Secondly, we take up methods of approximation for an operator f acting between infinitedimensional Banach spaces X, Y by a sequence of mappings (f,) acting between finitedimensional subspaces of X and Y . Finally, we consider infinite-dimensional homotopy theory of a mappingf E C ( X , Y ) . In this theory one attempts to answer questions concerning the mapping properties off by homotopy, i.e., by continuously deformingf to a simpleri for which the mapping property can be easily determined. This last theory leads to the association of various numerical topological invariants withf(in many cases). We end the chapter by applying the invariants to a variety of specific problems. In Chapter 6, we take up additional methods that can be used for gradient mappings. This distinction is well illustrated by the partial differential equations defining fluid flow described in Chapter 1. For problems involving ideal (i.e., inviscid) steady flow, the associated Euler equations generally define gradient mappings between appropriate Banach spaces. Indeed this fact will be utilized in the problem of global vortex rings (to be described in Section 6.4). However for steady viscous flow, the more general Navier-Stokes equations do not define a gradient mapping and so to study problems concerning such flows, methods of study utilizing the theory of this Chapter, must be considered. Moreover, in Section 5.5, we show the power of these methods by solving a classic problem of periodic ideal fluid flow.

5.1 Linearizatlon

’

Let f ( x ) be a C operator defined on a domain D of a Banach space X with range in a Banach space Y , i.e., f E C ‘ ( D , Y ) . In Part 11, various local properties of the operatorf(x) near a point xo E D were derived from the behavior of the Frechet derivativef’(x,). Here it is natural to piece this information together and consider those global properties of the mapping f ( x ) that can be determined from the behavior of f ’ ( x ) at each point x E

D. 217

218

5 GLOBAL THEORIES FOR GENERAL NONLINEAR OPERATORS

Simple examples show that even if the Frechet derivativef’(x) is a linear homeomorphism or surjective (as a linear map from X to Y ) for each x E D, f(x) may not share these properties. In this section we clarify this situation by the use of simple topological considerations based on ideas concerning covering spaces in the case of homeomorphisms and connectivity properties of f(D) for surjectivity.

5.1A Global homeomorphisms

Suppose that the Frechet derivativef’(x) of an operator f E C1(X, Y ) is a linear homeomorphism from X to Y for each x E X . Then the inverse function theorem of Chapter 3 implies f is a homeomorphism of a neighborhood Ux of each point x onto f( U,). Thus we inquire: What are the properties that f must possess in order for f to be a homeomorphism of X onto Y ? A topological approach to this question that proves to be quite natural in this context is the notion of covering space. Indeed from the well-known definition and properties that we summarize below, it follows easily that a necessary and sufficient condition for f to be a homeomorphism of X onto Y is that ( X , f)be a covering space for Y. This follows from the simple connectivity of the Banach space Y . Definition Suppose X is a connected, locally connected Hausdorf topological space. (X,f)is a covering space of a connected Hausdorf topological space Y if:

(i) f is a continuous mapping of X onto Y , and (ii) every y E Y has an open neighborhood Uy about it such that f - I ( Uy)is the disjoint union of open sets 0;in X each of which is mapped homeomorphically onto U by f. The following properties of covering spaces will be useful in the sequel. Indeed it is these properties that are crucial for applications to analysis. covers Y , then the covering map f has the properties: Suppose (X,f) (i) Unique path lifting Suppose f(xo) = y o and L ( t ) is a continuous path in Y with L(0) = y o , then there is one and only one continuous path p ( t ) in X withp(0) = xo and f p ( t ) = L ( t ) for t E [0, I]. (ii) Covering homotopy property Suppose L , ( t ) and L,(t) are continuous paths in Y with fixed base point that are homotopic, then these paths can be lifted to continuous paths e,(t)and c,(t) in X that are also homotopic with fixed base point. (iii) For each y E Y , the number of points in f -I(y) is constant. (iv) Suppose Y is simply connected, then f is a homeomorphism.

5.1

219

LINEARIZATION

For the proofs of these results we refer the reader to the book by Hu (1959). See also Spanier (1966). Thus to answer the homeomorphism problem we shall determine necessary and sufficient conditions for a homeomorphism to be a covering map. To this end we prove (5.1.1) Suppose D is a domain of X andf E C ( D , Y ) , then necessary and sufficient conditions for the pair (D, f)to cover f(0)are: (i) f is a local homeomorphism; and (ii) f lifts line segments, i.e., for any finite line segment L ( t ) E f ( D ) joining yo = f ( x o ) and E f ( 0 ) for any xo E D , there is a curve ~ ( tsuch ) thatf(x(t)) = L ( t ) with x(0) = x,,. (See Fig. 5.1.)

):,

FIG. 5.1

Finding a curve x ( f ) , lifting the line segment ( L( r) ,0 G r

< 1).

Proof: (R. Plastock) The necessity of (i) and (ii) follows from the results mentioned in the above discussion of covering spaces. To prove the sufficiency of (i) and (ii), we proceed as follows. Sincef is a local homeomorphism, f is an open mapping and possesses the unique lifting property for line segments. Hence if y E f ( D ) and { x a } =f-'(y), then there is an open ball B ( y , r ) = ( z 1 llz - yII < r , z E Y } contained in f ( D ) and the set of curves emanating from xu,

ox,= {x&) I t E [O,

11,

E X,

x,(O) = xu, f ( x a ( t ) ) =

y + try}

220


is well defined for any Z of norm 1. (The curve x,(t) exists by hypothesis (ii).) Now we show that the sets 0, are disjoint open sets mapped homeomorphically onto B ( y , r), and f - ' ( B ( y , r ) ) = U ,Ox,. Clearly once these facts are established, we have shown that (D,f) covers f ( D ) . By ,O, = f - ' ( B ( y , r)). construction, it will also follow that Before proceeding further with the sufficiency half of the proof, we make the following observations, assuming f satisfies (i) and (ii).

u

(a) Distinct paths in Ox are mapped onto distinct radii in B ( y , r ) . (b) If Pl(r) and P 2 ( t ) are any two paths that intersect,then either P , and P , are mapped onto the same radius, or the point of intersection can only occur at t = 0. (c) If f : D + f ( D ) is a local homeomorphism with M I and M , open subsets o f ' D with nonempty intersection on each of which f is a homeomorphism, then f is a homeomorphism of M , u M , onto f ( M 2 ) provided f ( M , ) n f ( M , ) is connected; thus if f(M,) and f ( M 2 ) are balls, f(M,) n f ( M , ) is convex and thus connected. From the construction of Ox and (a) above, we can conclude that each 0, is mapped onto B ( y , r ) in a one-to-one way, and thus homeomorphically since f itself is an open map. I t remains to show that the Ox are disjoint open sets whose union is f - ' ( B ( y , r ) ) . The disjointness follows from (b). For if P l ( t l )= P2(t2)= X E Ox,n Ox2.(x, # x,), then showing that P,(O) = P,(O) will be the desired contradiction. However, if PI and P, are mapped onto the same radius, then necessarily t , = t,. Thus s = { t I P l ( t ) = P2(t), 0 < t < 1 ) is nonempty, open, and closed. Hence s = [0, 11 and in particular, P,(O) = P2(0). Next we show that each Ox is open. Let u E Ox,f(u) = u, f ( p ( t ) ) = (1 - t)y tv and p ( 0 ) = x. By compactness we cover p ( t ) with a finite number of open sets D, each of whose image under f is a ball. Then by (c) above, A = UjDjis mapped homeomorphically onto f(A). Now we claim that there is some number E > 0 such that if 111, - w I I < E the line joining y and w €f(A). Indeed otherwise there are sequences w,+v such that y ( f n= ) (1 - tn)y t,w, Ef(A), so that y(t,)+(l - 7)y + 7v €f(A). But this is a contradiction since f(A) is an open set. Consequently, by restricting f to A, we observe that f-'lA ( w 1 IIw - u11 < c) is an open set in 0, containing u . Finally, we show that f - ' ( B ( y , r ) ) = U ?Ef-Ijy)Ox. Since it suffices to show the inclusion of the right-hand side in the left, we suppose x Ef-'(B(y, r)). Let L ( t ) = (1 - t ) f ( x ) y C B ( y , r ) . By hypothesis, there is a path P ( t ) such-that P ( 0 ) = x and f i P ( t ) ) = L(t). In particular, P(1) ~ f - ' ( y ) .If we let L ( t ) = L(l - t ) and P ( t ) = P(l - t ) , thenf(P(t)) = L"(t), P"(0) Ef-'(y), and & I ) = x. Thus x E OpCi). Thus the proof of (5.1.1) is complete.

+

+

+

Thus to show that the C ' mapping f : X -+ Y is a global homeo-

5.1

22 1

LINEARIZATION

morphism, we need only prove that f is a local homeomorphism and for any y E Y and xo E X there is a curve x ( t ) E X such that (5.1.2)

f ( x ( t ) ) = t.r.

+ (1 - f)yo,

x(0) = xo,

for t

E

[0, 11.

(This requires that we know the surjectivity off a priori.) In this sense then the homeomorphism question is reduced to a simpler one-dimensional problem. A useful explicit method for constructing the curve x ( t ) satisfying (5.1.2) is based on the theory of ordinary differential equations in Banach spaces. Indeed, by differentiating the relation (5.1.2) with respect to t , we find that the curve x ( t ) satisfies the initial value problem (5.1.3)

dx/dt = [ f ’ ( x ) ] - I ( y -yo),

x(0) = xo.

Conversely, if (5.1.3) has a solution x ( f ) that exists for t

E [0,

I]. the curve

x ( t ) will satisfy (5.1.2). This idea is useful for finite-dimensional problems

due to Peano’s theorem (3.1.28). For infinite-dimensional problems, the study of (5.1.3) is less useful. The argument based on ordinary differential equations however, can easily be abstracted. This is clearly seen in the proof of the following result of Banach and Mazur. (5.1.4) Theorem (i) Let f E C ( X , Y ) . Then f is a homeomorphism of X onto Y if and only iff is a local homeomorphism and a proper mapping. (ii) Provided f E C ‘ ( X , Y ) ,f is a diffeomorphism if and only i f f is proper andf’(x) is a linear homeomorphism for each x E X .

Proof: (i) If f E C ( X , Y ) , the necessity of the conditions stated are immediate. To demonstrate the sufficiency, we first note that f maps X onto Y . Indeed, since f is a local homeomorphism, f ( X ) is an open set; while the fact that f is proper implies that f ( X ) is closed. Consequently, the connectedness of Y implies that f ( X ) = Y . Thus by (5.I.l), we need only demonstrate the existence of a curve x ( t ) satisfying (5.1.2). The fact thatf is a local homeomorphism, implies that for some small E > 0 and r E [0, E), there is a curve x ( t ) satisfying f ( x ( t ) )= 9 (1 - ‘)yo. Let p > 0 be the largest number for which x ( t ) can be continuously extended to satisfy f ( x ( t ) ) = ty + (1 - t)>*,, for 0 Q t < p, and suppose t , + p . Since L = { y ( t )1 y ( t ) = ty + (1 - ‘)yo, t E [0, l]} is compact and f is proper, f - ’ ( L ) is compact, so that x(ti) has a convergent subsequence x ( t i n ) + X , as tin + p. By continuity, f(F) = fiy + ( 1 - ,8)yo.By virtue of the fact that f is a local homeomorphism, x ( r ) can be continuously extended for t > p. This contradicts the maximality of p, and we conclude that x ( t ) exists for t E [O, I] independently of xo E X and y E Y . (ii): If f E C1(X, Y ) is a diffeomorphism with inverse g, then f ’ ( x ) is

+

222

5 GLOBAL THEORlES FOR GENERAL NONLINEAR OPERATORS

certainly a linear homeomorphism for each x E X since the relations = x can be differentiated. Thus the necessity of the conditions stated is clear. To demonstrate the sufficiency of the conditions, we use the inverse function theorem and (i) above to prove that f is a homeomorphism, with inverse g (say). Then since f g ( y ) = y for each y E Y , the differentiability off implies the differentiability of g . The following quantitative criterion is due to Hadamard, for the case of finite dimensions.

fg(y) = y , &(x)

(Hadamard) Suppose f E C ' ( X , Y ) is a local homeomorphism and { ( R ) = i n f l l x l l G ( lR/ ~ ~ [ f ' ( x ) ] - ' ~Then ~ ) . if J " S ( R ) dR = 00, f is a homeomorphism of X onto Y . In particular, if ll[f'(x)]-'[l 6 M for all x E X,f is a homeomorphism of X onto Y . (5.1.5)

Proof: We shall prove that ( X , f ) covers f ( X ) , and in addition that f(X) = Y . By (5.1.1), ( X , f ) will cover f(X) if and only if f lifts line segments. To establish this fact we argue as in the proof of (5.1.5). Let xo E f -'(yo), and y E Y , then we seek a curve x ( t ) such that f ( x ( t ) ) =y (1 - t)yo. Sincef(x) is a local homeomorphism, x ( t ) exists for small t. Let /3 be the largest number such that x ( t ) can be continuously extended to 0 6 t < /3 satisfying f ( x ( t ) ) = fy + (1 - t)yo. We shall prove that lim,+D x ( t ) exists and is finite. Assuming this fact for the moment, then as in the proof of (5.1.4), x ( t ) exists for t E [0, 11, so that f lifts line segments. Since y E Y is arbitrary, this argument shows that f(X) = Y . Consequently, we need only use the hypotheses of the theorem to prove that limr+Bx ( t ) exists and is finite. If ll[f'(x)]-'\l 6 M , this fact can be easily established. Indeed, if t < /3, x ( t ) satisfies the equation

+

(*I

x ' ( 4 = [f'(x(t))l-'(.Y -Yo)?

so that for any t , , t ,

< p,

Thus x ( t ) satisfies a Lipschitz condition for t < /3 and since X is complete, lim,+D x ( t ) axists and is finite. More generally, if J " { ( t ) = 00, the argument just given can be modified as follows. We can define the length of x ( t ) for 0 6 t < /3 with respect to '~~, the weight g(x) = l / ~ ~ [ f ' ( x ) ] -as L,(x(t),

[Ol

PI) = J0~ ~ ( x ~ ~ ) ) l I x Ydt. t)ll

Now we reach the desired conclusion by proving that if J " S ( t ) = 00, the metric defined on X as above is complete in the sense that if L,(x(t),

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223

LINEARIZATION

[0, p)) < 00, then h d f x (l t ) exists and is finite. Let 0 < s above estimates, since dllx(t)ll is of bounded variation,

< p.

By our

Consequently, since J " { ( t ) dt = 00, Ilx(t)l\ is uniformly bounded for t E [0, p). On the other hand, since { ( t ) is nonincreasing and J;{(t) dr = M, sup{t I { ( t ) > 0 ) = 00, so that g(x) is bounded from below on any bounded set. In particular, g ( x ( t ) ) is bounded from below for 0 < t < p, say IIg(x(t))ll > G . Now let f,TP, then n

n

(**I

c lIx(t,+1)

- x(t,)ll G

r=l

c

i s 1

SUP llx'(~)Il(t,+,- t,) re[r,.r,+,]

Thus (**) implies that ~ ~ = l l l x ( t ,+ Ix(t,)JJ ) < M, so that {x(t,)> is a Cauchy sequence in X . Consequently lim,r+p x ( t ) exists and is finite. As an application of (5.1.4), we prove

(5.1.6) Suppose that f ( x ) is a continuous mapping from a reflexive Banach space X into its conjugate space X * with the property:

(t)

(f(4-f(Yh x - A

2 q(11x -Yll)llx -YII.

where q ( r ) is a positive function satisfying q(0) = 0; and q ( r ) + co as r + M. Then f is a homeomorphism of X into X * . Proof: We prove that f is proper and a local homeomorphism. To show that f is proper, we note that (t)implies that whenever x, + x weakly and f ( x , ) + z, then f ( x ) = z and x, + x strongly. Consequently, suppose ( f ( x , ) } is convergent in X * , then by (t)and the reflexivity of X , {x,} is bounded, and after possibly passing to a subsequence, we may suppose that x, + X weakly in X . So { x,} is strongly convergent, by property (t). To prove that f is a local homeomorphism, we see by virtue of (t)that f maps X injectively onto its range. In addition, if x I = f - ' ( z , ) and x2 =f-'(z2), then (t)also implies that llzl - zzll 2 q(llf-'(zl> -f-'(z2)l1), so thatf-I is continuous. Thus by (5.1.4),f is a global homeomorphism of X onto Y .

In practice, the application of Hadamard's theorem often requires an initial decomposition of the appropriate Banach spaces X , Y as well as the

224


utilization of special properties of the mappingf relative to this decomposition. Such an example occurs in the differential geometric problem of finding a Riemannian metric of constant Gaussian curvature - 1 on a smooth two-dimensional Riemannian manifold (%, g) of negative EulerThis . problem was mentioned in Section Poincare characteristic x( ”X) 1.1A in connection with the uniformization problem for algebraic curves. By virtue of our discussion there (cf. the equation (1.1.5)), the existence of ‘ 7such a metric can be ensured by finding a smooth solution defined on 3 for the following partial differential equation (5.1.7)

Au - K ( x ) - e2” = 0.

In this connection we shall use Hadamard’s theorem to prove (5.1.8) Theorem A necessary and sufficient condition for the solvability of (5.1.7) is that ( , m K ( x ) dVg < 0. Consequently if K ( x ) denotes the , and x(%) < 0, (5.1.7) is always solvable, Gaussian curvature of ( “ X g) by virtue of the Gauss-Bonnet theorem. Proof: The proof requires three steps. First, we obtain a reformulation of the problem into an appropriate operator equation defined on the Sobolev space W l ,2 ( % , g) carried out in such a way that any (generalized) solution u to the operator equation is automatically a smooth solution of (5.1.7). Secondly, in order to apply Hadamard’s theorem we must decompose the resulting abstract operator equation relative to the facts that the g) consisting of the constant Laplace operator A has a kernel on functions; and in fact the desired conclusion itself asserts that the associated mapping f, unless modified, certainly will not be a global homeomorphism. The final step consists in estimating the size of the Frechet derivative of the appropriately modified f in such a way as to satisfy the hypotheses of Hadamard’s theorem (5.1 S ) .

(a,

Step 1: The desired reformulation of (5.1.7) is easy in this case, provided we use the duality method of Section 2.2D. Indeed defining the operators L and N implicitly by the formulas

we see that L is a bounded self-adjoint mapping of the Sobolev space W , , , ( % , g) into itself, while by virtue of the estimate (1.4.6) N is a C 1 mapping acting between the same spaces. Consequently the partial differential equation (5.1.7) can be written in the form (*)

Lu

+ Nu =f

where

(f, u ) = - J K ( x ) u .

The verification that solutions of this operator equation in W1,*(Gx , g) are

5.1

225

LINEARIZATION

automatically smooth enough (possibly redefined on a set of Lebesgue measure zero) to satisfy (5.1.7) pointwise is a consequence of the Lp regularity theory described in Section 1.5B and the estimate (1.4.6). Step 2: We now decompose the operator equation (*) by setting W l ,2(E)TL, g) = H and writing H = Ker L @ H I and denote by P the canonical projection of H onto H I . Then if a tentative solution u = w c of (*) relative to this decomposition is given, (*) is equivalent to the pair

+

Lw

+ P N ( c + w ) = Pf;

e2ci,te2w =

-

This result follows since H I is the subspace of functions in H of mean value zero over ( L%, g). Now the second equation shows that the constant c is determined as a function of w if and only if Ju,RK< 0. Using this value c = c(w) in the first equation of the pair (**), we see that our problem reduces to showing the operator on the left-hand side of the first equation of (**) is a global homeomorphism of H I onto itself. Step 3: We conclude the proof by showjng that when regarded as a C ‘ mapping of H , inio itself the operator f ( w ) = L + P N ( c ( w ) + w ) has a Frechet derivative f ’ ( w ) that is an inv_ertible linear operator for all w E H I and in fact the Frechet derivative [ f ’ ( ~ ) is] uniformly ~ ~ bounded. Then Hadamard’s theorem (5.1.5) implies is a global homeomorphism o,f H I onto itself as required. To this end we must compute and estimate f’( w ) . This is most easily done, in the present case, by returning to the implicit definitions of L and N and estimating the following quadratic form (defined on H I ) , as follows with u E H .

fl

Note that on H I , the norm reduces to the-Dirichlet integral. Consequently, the Lax-Milgram lemma (1.3.21) iinpliesf’(w) E L ( H l , H I )is an invertible linear operator and Il[f.(w)]-’II < 1, which is the desired uniform bound. Thus our result is established. 5.1 B

Mappings with singular values

Our results on global homeomorphisms, just obtained, may still be useful even though a C I operator f has singular values. Indeed, f may be a global homeomorphism when certain sets of finite codimension are excised from its range and domain. As we shall see this reduces the study of the mapping properties of f to a finite-dimensional problem. We illustrate this idea in the simplest case where the sets in question are linear subspaces. First, we present an abstract result and then apply it to

226


determine the exact structure of the range of a specific semilinear elliptic differential operator (supplemented with Dirichlet boundary conditions). (Additional applications of this fundamental abstract idea will be given in Section 5.3.) The abstract result we have in mind is embodied in (5.1.9) Reduction Lemma Let X and Y be Banach spaces, with L E L ( X , Y ) a Fredholm operator of nonnegative index p and N E C ’ ( X , Y ) . Moreover, suppose for some fixed number c 0, the following result holds off some finite-dimensional subspace W = Ker L @ V (i.e., for x E l):

>

w

(5.1.10)

Lx

+ PN’(u)x

is invertible and 11 L x

+ PN’(u)xll > ellxll,

where P is the canonical projection of Y onto L ( W L ) . Then f E.Range(L + N ) if and only if a certain system of dim W - p equations in dim W unknowns, defined by (5.1.12) below, is solvable and the solutions of L x N x = f correspond precisely to the solutions of (5.1.12). Moreover if L x + N x = f with x = wo + w,,w , E W , w I E W ’ , the following estimate holds

+

(5.1.11)

llwlll

< clllwoll + c2

(cl, c2 absolute constants)

provided N ’ ( x ) is uniformly bounded. Proof: We decompose X into the direct sum X = W @ W’ and Y = L( W ’) @ Yo with Po the canonical projection of Y onto Yo. Then for x = w o + w 1 and f E Y , the equation L x N x = f can be rewritten as the system (5.1.12) (i) Lw, PoNx = P , f , (ii) L w , + P N x = Pf.

+

+

Now regarded as a C ’ mapping of W L into L( W ’ ) , A w l = L w , + P N x has (by virtue of the inequalities (5.1.10)) a Frechet derivative A’(u)w, with lower bound I I A ’ ( ~ ) W I I I = IILW, + P”(u)w,II > ellw1ll. Moreover, by Banach’s theorem (1.3.20), since L E L( W I , L( W’)) is invertible, so is A ’(u). Consequently, Hadamard’s theorem (5.1.5) implies that (5.1.12 (ii)) can be uniquely solved by an element w , = w I ( w o ,Pf) that depends smoothly on wo and Pf. Moreover regarding w , parametrized by wo (for Pf fixed) we find by differentiating (5.1.12(ii)) with respect to wo in the direction u L w ; ( w , [ u ] ) + P N ’ ( x ) { u + w ; ( w o [ u ] ) }= 0 Consequently, a simple estimate using (5.1.10) and the uniform boundedness of N ’ ( x ) shows that IIw;(w,)[u]ll/llull is uniformly bounded for all u and so 4 I W l I l Q ~ollwoll+ K , ,

5.1

227

LINEARIZATION

where K , and KO are absolute constants. Now, this estimate implies the bound (5.1.1 1). Finally, we observe the system (5.1.12 (i)) can now be regarded as involving dim Yo equations in dim W unknowns since x can be written x = w o + wI(wo, PjJ Since L is a Fredholm operator of index p , we note that dim Yo = dim W - p , so the lemma is established. (5.1.11’) Moreover, if the operator N ( x ) is uniformly bounded, the function w = w ( x , , Pf) is also uniformly bounded as wo varies. Proof: We now suppose N is uniformly bounded. Then (5.1.12) implies

(*I ’ IILwll Q IIPf’ll + I I P N ( w o + w)ll Q K , where K is a positive absolute constant. Since L is invertible as a linear mapping between W and L ( W I). Thus, (*)implies that 11 wJI is uniformly bounded. Turning to a simple but informative application of the reduction lemma (in which an infinite-dimensional problem reduces to a one-dimensional one), we consider the mapping A defined by the explicit semilinear elliptic partial differential operator (5.1.13) Au E Au + f ( u ) , u laa =0 defined on a bounded domain Q E R N (augmented by null Dirichlet boundary conditions on Q). Here A denotes the Laplace operator relative to D with eigenvalues A , < A, Q A, with A , > 0, a n d f i s a C 2 strictly convex function satisfying f(0) = 0 and the asymptotic relations (5.1.13’)

0 < lim /--W

f’(l)

0 for all u E H. Moreover N is a bounded mapping of H into itself with a uniformly bounded Frechet derivative N ’ ( u ) defined implicitly by the relation ( N ’ ( u ) w , u) = lo(f’(u) - X,)wu. The verification of these properties is routine. Actually, N defined as above is a C ’ mapping (and this fact will be useful in the sequel). To verify this fact we observe that the asymptotic properties (5.1.13’) imply thatf”(t)-0 as 1+w, so that f “ ( t ) is uniformly bounded over (- 00, w). Moreover if u,, + u in H , f ’ ( u , ) +p’(u) in measure over Q. Consequently an application of the Lebesgue dominated convergence theorem shows that N ’ ( u ) calculated directly from the implicit definition of N exists and is continuous in u. Step 2: (Reduction to a one-dimensional problem): We apply the reduction lemma to the operatorf= L - N. In the present case p = 0 since L is self-adjoint. Moreover, in the present case the subspace W in the statement of the reduction lemma coincides with the onedimensional subspace Ker L = ( u I Au X,u = 0,u € H ) . (Here we have used the fact that

+

5.1

229

LINEARIZATION

the lowest eigenvalue of A on 52 is always simple.) Indeed, for 01Ker L, by (5.1.13) and the properties of

Thus, defining the norm in H by setting i i u i l ~= lo] V uI2 and by applying (1.3.20), we find that W can be chosen to be Ker L. Consequently, according to the reduction lemma it w ( f l g l ) ) = -Jngu,, where u , is suffices to study the one-dimensional problem - P , N ( t u , the positive normalized eigenfunction of A associated with D. Denoting the negatives of either sides of the above by h ( r ) and K respectively, we need only study the solutions of the equation

+

Moreover from (5.1.1 1) the following estimate holds for Ilw(t, g,)/I < c I f + c2. Step 3: We now establish the results (a)-(d) mentioned in the statement of (5.1.14), by proving the following two properties of the function h ( t ) defined by (5.1.15): (a)h(t)-+ M as It1 + M and ( p ) inf, h ( t ) > - m is the only critical point of h ( t ) and is attained exactly once at t = t , (say). Moreover t , is uniquely determined by g, and is a continuous function of g,. Assuming (a) and ( p ) for the time being, we observe that (a)-(d) as stated in (5.1.14) are immediate consequences upon setting a( g,) = h ( t , ) . Thus to demonstrate that singular values A ( S ) of A are precisely the points of the form g = h ( t , ) u , + g, as g varies over (Ker I,)’, we first differentiate the equation A ( t u , + w ( t , g , ) ) = h ( t ) u , + g , with respect to t and set f = t , , obtaining A ‘ ( f l u , + w ( f l , g,)) (u,+ w‘(f,)) = 0 so f l u l + w ( t l , glO) E S . Conversely, suppose u E S and A ( u ) = c l u l + g, E A ( S ) . So, by our results so far, u = t u , w ( t , g,) and for some u = aul + w , , A ’ ( u ) u = 0. Thus if P denotes the projection of H onto (Ker L ) I = H I , PA’(u)(au,) = - P A ’ ( u ) w , ; and since - PA’(u) is invertible on H I ,

+

w , = -(PA’(u))-’PA’(u)u, =

Thus w = au,+ aw’(t,gl) and since ( I

-

W’(t,,

g,).

P ) A ’ ( u ) u = 0 implies h ’ ( f )= 0, we find t = f , by

( p ) ; so u = t l u l + w ( t l , g , ) as required. Finally, we prove (a)and ( 8 ) .The basic problem is the justification of the simple idea that

both facts are easily obtained if the contributions to h ( t ) due to w ( t l , g , ) are neglected. Given a subset ( f ) with ( t ( + 0 0 , we note that it suffices to show h ( t ) + + M for any two sequences t = r, M and t = s, + - 00. This will follow by finding an absolute constant c , > 0 such that

+

We demonstrate (*) here since (**) is completely analogous. To this end, we note that the a priori estimate mentioned at the end of step 2 implies that for t sufficiently large ( w ( t ) / t ) is uniformly bounded and so can be assumed to possess a weakly convergent subsequence ( w ( r m ) / f n )that without loss of generality can be assumed to be strongly convergent in L,(D) and pointwise convergent (a.e.) to an element iii E H . Next partitioning D into three sets D + , Q - , and Do according as u I ( x ) w ( x ) is positive, = A, - e + and lim,-,+mf’(f) negative, or zero, we find from (5.1.15) that setting lim,,,f(t)

+

= A , - €-,

230


Here we have used the Lebesgue bounded convergence theorem and the observation that the integral of f ( t n u l w ( t n ) ) / t n over 0, also tends to zero as n+m. Moreover, lQwuI= 0 since this result is preserved under weak convergence in H.Consequently

+

Thus we can estimate the right-hand side of (5.1.16) from below by changing the coefficient of the negative integral from A, - c - to A, - c + . Thus combining (5.1.16) with the equality displayed above we find that, for c , = A, - A, - c + > 0, since w l u , in the L, sense, lim f“+W

-1,

Thus the desired result (a)is established by making strong use of the asymptotic linearity off to invoke the orthogonality of u, and w ( f ) . The proof of ( p ) is somewhat more intricate utilizing, as we shall see, the convexity of f strongly since the asymptotic linearity off defined by the relations (5.1.13) cannot be used. We show below that at a critical point to the function h ( t ) has the property

in case the function h ( t ) is twice differentiable. Then we use the positivity of the eigenfunction u, to prove sgn[u,+ w’(to)]>O, so that h”(t,)>O. To this end we rewrite (5.1.15) in the form h ( t ) = P A ( u ( t ) ) , where P is the projection of H onto Ker L. Then a formal calculation, assuming h’(ro)=O, yields the formula h”(to)=(A”(u(to) (u‘(ro), u’(to)), u’(to)) and consequently the representation (*). Moreover u’(t)= u , w ‘ ( t ) satisfies the linear equation A’(u(r))u’(t)=O which in the present case implies u = u’(f) is an eigenfunction of the equation

+

(5.1.17)

AU +f’(u(r))o = 0,

“Ian

=0

corresponding to the eigenvalue A = 1. Now the asymptotic relations satisfied byf’(s) and the extremal characterization of eigenvalues imply A = 1 is the smallest eigenvalue of (5.1.17) and moreover sgn u’(t) = sgn(u, + w’(t)) is constant in 0. Now since l a u l w ’ ( r ) = 0, w ’ ( t ) > 0 for some open subset Q‘ of 0.Thus sgn(u, + w’(t)) > 0 over 0, as desired; and ( p ) is proven in case the function h ( t ) is twice differentiable. However, in general, h ( t ) is nor C2, so a modification of the direct idea just given is necessary. To this end we note it suffices that for It - tOl sufficiently small, h ( t ) > h(ro), or equivalently that sgn h’(t) = sgn(t - to). To establish this we use the defining relation A’(u(t))u’(r)= 0, to find

J n f ’ ( ~ ( t ) ) u Y t ) w Y t o )= j n f ’ ( U ( l o ) U ) ( t ) W r ( t g ) ) . Consequently a short computation gives the relation (5.1.18)

h’(t) - h’(t0) = L { f ( u ( t ) ) - f ’ ( u ( f o ) ) ) U ’ ( t ) d ( t O ) .

Now instead of computing h”(t,,) directly we divide the above integral on the right into two parts an integral over Q , = (xlu’(to) < 1) and an integral over 0 -0,. On 0, an easy application of the Lebesgue dominated convergence theorem to (5.1.18) shows that since f” is uniformly bounded, limr-tfo(h’(t) - h ’ ( Q ) / ( t - r& exists and coincides with the right-hand side of (*), and thus is positive. Consequently it remains to discuss (5.1.18) on Q - Q , . Since u(r)+u(ro) in L2(0),we first note that any sequence t n + t o has a subsequence (which we again relabel r,) such that for n > no and some z > 0 u’(tn) > f (a.e.) and (u(t,) - u ( r o ) ) / ( t n - to) 2 L. Thus for n

> no on 1;2 - Q , , f ( u ( f , ) ) > f(u(ro)) and

u’(tn) > 0,

so the integral in (5.1.18) on

5.2 FINITE-DIMENSIONAL APPROXIMATIONS

23 1

Cl - 0,is positive for f > to, and the desired result is established. (See Note B at the end of the chapter for extensions of (5.1.14).)

5.2 Finite-DimensionalApproximations

Properties of solutions of an operator equation f(x) = 0 defined on a Banach space X can be studied by approximating both f and X by sequences of mappings {f,,)and spaces { X , , } . Assuming that the pair (f,,, X,,) converges to (f, X ) in a well-defined sense, one attempts to analyze the solutions { x,,) of f n ( x n ) = 0 on X,, as n -+ 00 with a view to showing that appropriate subsequences of { x n } converge to solutions of f(x) = 0 on X . Assuming dim X = 00, we shall discuss this circle of ideas by approximating X by a sequence of finite-dimensional subspaces X,, and f by a sequence { f,,)(with finite-dimensional range) obtained by restricting the domain off to X,,. 5.2A

Galerkin approximations

More precisely, let X be a real reflexive separable Banach space (with conjugate space X * ) and let { X,,) be a fixed sequence of finite-dimensional subspaces of X such that X,, c X,,, I and U"=, X,, is dense in X . To fix notation, let P,, denote the projection of X onto X,,, P,* denote the conjugate operator, and X i = P,*X*. Thus if f is a bounded continuous mapping of X into X * and g E X * , then a sequence of finite-dimensional approximations to the equation f ( x ) = g defined on X can be denoted (5.2.1),,

P,*f(x) = P:g,

x E X,,.

The system (5.2.1),, is often referred to as the Galerkin approximation for the equationf(x) = g. Here we have limited attention to equations defined on reflexive Banach spaces X in order to use the weak compactness of bounded sets in X . Indeed, in order to deduce the solvability of f(x) = g on X from the solvability of (5.2.1),, for n sufficiently large, we need only: (i) determine an a priori bound for the solutions {x,,) of (5.2.1),,, llx,,llx Q M (say), where M is independent of n, so that (after possibly passing to a subsequence) one can assume that {x,,) is weakly convergent with unique weak limit X; (ii) use the qualitative properties o f f and the Galerkin construction itself to show that f(X) = g. Thus (in the simplest case) if f is weakly sequentially continuous as a mapping from X to X * , (i) immediately implies that f(X) = g. Actually a closer study of the approximation procedure (5.2.1),, allows

232


this weak continuity assumption just mentioned to be considerably reduced. To this end, we observe that two characteristic qualitative properties of the solutions { x,) of the Galerkin approximations can be expressed by the two sequences of equations (5.2.2)

(a) (f(xn), xn> = ( g , Xn);

(b) (f(X,), 2) = ( g . Z ) ? 2.E X,. These results are obtained by taking inner products of (5,2.1),, with xn and z , respectively. Thus for convergence purposes, once (i) is established, we can suppose not only that x,-+ X weakly, but also that f ( x , ) + E weakly (since f is bounded). Furthermore, (5.2.2b) implies that E = g, while (5.2.2a) yields the fact that (f(xn), x,) + ( g , X). Hence the crucial property of the operator f necessary to establish (ii) from (i) can be expressed as Condltlon (G) If x,+x weakly in X , f ( x , ) + y (f(Xn), Xn) + ( Y , x), then f(x> = Y *

weakly in X * , and

This condition is satisfied by all continuous mappings between finitedimensional Banach spaces, all weakly sequentially continuous mappings, and, as we shall see in the sequel, a large class of mappings associated with quasilinear elliptic partial differential equations. We are now in a position to prove the following: (5.2.3) Theorem Suppose f is a bounded mapping of a real, reflexive separable Banach space X into its conjugate space X * satisfying the conditions

(I) (11)

(f(x), x)llxll-'+ 00 as IIxlI f satisfies Condition (G).

--j

00,

and

Thenf is a surjective mapping of X onto X * , and for any g E X* a solution of f(x) = g can be obtained as the weak limit of an appropriate subsequence of the solutions of the Galerkin approximations (5.2. I),. Proof: The main idea is to use condition (I) to ensure both the existence of a solution x, of (5.2.1),, and a uniform a priori bound for the sequence {x,). Then, as mentioned above, (11) is used to show that a weakly convergent subsequence of { x,} converges to a solution of f ( x ) = g for any g E X*. Now suppose that g is an arbitrary element of X*. Then to show that (5.2. l), is solvable for each n, we show that the finite-dimensional mapping f, = P,*f, regarded as a mapping from X , + P,*X,, is surjective. To this end, we observe that for x E X,, hypothesis (I) implies IIxIIin'(fn(x)9

X)

= (j(x>,

X)IIXII~'

-+

00

as

IIXII,~,+

00.


233

Thus the Brouwer degree of f, at any fixed element of P,*X, relative to a sufficiently large sphere { x I llxll < R } is unity, so that f , ( x ) = g has a solution in X,. Consequently, there is an x, E X, satisfying (5.2.l),. Futhermore, by (5.2.2a) and Schwarz's inequality, ( f f l ( x f l )xfl> . = (f(xfl).xfl>= (gq

Xn)

II gll

IIxflII.

Thus hypothesis (I) implies that the sequence {x,} is uniformly bounded. Hence after passing to suitable subsequences, we may suppose that { x,} converges weakly to X in X , and, by virtue of the boundedness off, that f ( x , ) - + g weakly in X*. Furthermore, repeating the argument given after (5.2.2), we may also suppose that ( f ( x , , ) , x,) --+ (g, X). Then, since f satisfies Condition (G), f(X) = g. Since g was arbitrary, f is surjective and the theorem is established. The applicability of the result just obtained will now be clarified by (i) determining a large class of mappings f satisfying Condition (G) and (ii) demonstrating the necessity of some hypothesis such as Condition (G) for the convergence of Galerkin approximations.

(5.2.4) Example (Theorem (5.2.3) is false if hypothesis (11) is removed.) Let X = I,, the Hilbert space of square summable sequences, and let an element x E I , be written as x = (x,, x,, . . . ) with ljxll' =x,"=,x,?. If T x = (dl - /lx1I2 , x,, -s2,. . . ) for iixll 1, then T is a continuous mapping of 12+ I , and ( 1 T,ll = 1 for all x E I,. Settingf(x) = x - Tx, we observe that f is continuous and ( f ( . ~ ) x) , = llxli2 - ( T x , x) > jlx1/2- 11 Txll IIxII. Thus

( f ( x ) , x)/llxll > IIx(/- I as llxil.+ 00. On the other hand, f is not surjective since f ( x ) = 0 has no solution. Indeed, if f(y) = 0, then since 11 Tyll = 1 . llyll = 1. Thus ify = ( y , . y , , . . . ) a n d y = Ty. theny, = 0 a n d y i + , = y i for all i. Hencey = 0, which contradicts the fact that llyll = 1.

We now determine a large class of mappings that satisfy Condition (G), but are not necessarily weakly sequentially continuous. To this end, we prove

(5.2.5) The following classes of mappings satisfy Condition (G): (i) continuous monotone mappings T of X+X* since ( T ( x ) T ( y ) ,x - y ) > 0 for all .x,y E X ; (ii) completely continuous perturbations of monotone mappings; (iii) mappings of the form T ( x ) = P x + R x : X + X*, where the mappings T , P, and R can be written in the form Tx = T ( x , x) with T ( x , y ) = P ( x , y ) + R ( x , y ) : X X X + X* and satisfy: (a) ( y - z , P ( x , y ) - P ( x , z ) ) 2 0.

234


(b) If x , + x weakly and ( P ( x , , x,) - P(x,, x), x, - x)+O, then R x , + R x weakly. (c) If { x , } and { y,} are weakly convergent in X with limn+my, = 0, then (Rx,, y,) +0. (d) For fixed x E X , R ( y , x ) and P(y, x) are completely continuous mappings of X + X * . (e) For fixed y E X the mappings P ( y , x) and R ( y , x) are bounded and continuous from the strong topology of X into the weak topology of X * , uniformly on bounded sets of the alternate variable. Proof: (i): Suppose T is monotone and w is an arbitrary element of X . Then for each n (5.2.6)

(x, -

W,

Tx, - T w ) 2 0.

If x , + x, Tx, -+y weakly, and ( T x , , x,) +( y , x), then by letting n + co in (5.2.6), we find (5.2.7)

(X

- W,Y - Tw)

> 0.

Setting w = x - A t in (5.2.7) for h > 0 and arbitrary z E X , we find ( z ,~ T ( x - Az)) > 0. Consequently, letting h+O, we see that ( z ,y - T x ) 2 0 for arbitrary z E X . Thus T x = y , as required. (ii):

A completely continuous perturbation R of a monotone mapping

P satisfies the hypotheses of (iii) with P ( x , y ) = P ( x ) and R ( x , y ) = R ( y ) . Therefore (ii) will follow from the more general case. (iii): The argument used is an extension of the proof in (i). For arbitrary w E X and each n. by hypothesis (a) (5.2.8)

(x, -

W,

T ( x , , x,,) - T ( x , , w ) ) > (x, - W , R ( x , , x,) - R ( x , , w ) ) .

Set R ( x , x) = R ( x ) and P ( x , x) = P ( x ) for any x E X . Then if both x , + x and Tx, +y weakly, while (Tx,, x,) +( y , x), by letting n + co and utilizing hypotheses (c) and (e), we find that (x, - x, Tx, - Tx)+O and ( x , - x, Rx, - Rx)+O. Subtracting, we obtain ( P x , - P x , x, - x ) + O , so that by hypothesis (d), (x,, - x, P ( x , , x,) - P(x,,, x))+O. Therefore by hypothesis (b), R x , + R x weakly; whence by hypothesis (c), (Rx,, x,,) + ( R ( x ) , x). As n + co in (5.2.Q and by virtue of hypothesis (d), we see that for arbitrary w (5.2.9)

(X

- W , Y - T ( x , w ) ) 2 (X -

W,

R ( x , X) - R ( x , w ) ) .

Setting w = x - Az in (5.2.9) for X > 0, we find, as in (i), after dividing by h and letting X-0, that ( t , y - T ( x , x)) 2 0. Thus y = T x , as required.

FINITE-DIMENSIONAL APPROXIMATIONS

5.2

235

5.28 Application to quasilinear elliptic equations

Operators satisfying Condition (G) arise naturally in the study of general quasilinear elliptic differential operators of divergence form. Indeed, let the differential operator Au =

2

14

( - 1)'*' D"A,(x, u,

. . . , Dmu)

m

be defined on a bounded domain D c R N . Then, as in Section 2.2 (iii), assuming that the coeificients A , ( x , u, . . . , D m u ) satisfy mild continuity and growth restrictions, we can associate with A an abstract operator T : W,,,p(C2)+ W_,.,(Zn) defined implicitly by ( T u , 'p)=

JA,(x, u , . Iulsm

. . , D"u)D*'p.

Q

Heuristically speaking, the ellipticity of A is specified by the dependence of A on the terms involving derivatives of order 2m, while the terms of A involving lower order derivatives can be considered as "compact" perturbations. Thus it is natural to separate the dependence of Tu on derivatives of u of order m from its dependence on the derivatives of u of smaller order. To this end, we write Tu as the sum of a principal part Pu and a remainder Ru, and in addition define (5.2.10)

( P ( u ,G),

'p) =

2

/ A , ( x , u,

. . . , D m - ' u , Dmo)D"'p,

I*I=m

(5.2.11)

( R ( u , G ) , q) =

x P.(x, . . . u,

, D m - I u , Dmr;)Da'p.

/a\ <m-I

+

Then, formally at least, T ( u , u ) = P ( u , u ) R ( u , u), where T ( u , G ) = P ( u , u) + R ( u , c), P ( u , 1 4 ) = Pu, etc. With the above paragraph as motivation, we can now interpret the hypothesis (a)-(e) of (5.2.5(iii)) in terms of the concrete differential operator A . Hypothesis (a) is an expression of the ellipticity of A , whereas (d) is an abstract statement of the fact that terms of order less than m are compact perturbations of A . Hypothesis (e), as mentioned above, imposes growth and smoothness restrictions on the coefficients A , of A . Hypotheses (b) and (c) require slightly more explanation. Equation (5.2.1 1) and Holder's inequality imply that for suitable conjugate exponents pa, q, (to be chosen) (5.2.12)

l(R(un)vyn)l

x

lulsm-l

llA,(x9 uf13. . . Dm~fl)llg,,llD~flIIp,.

236

5


Now if y,+O weakly in Gm,,(Q), then for la1 < rn - 1, D"Y,+O strongly in L,,(Q), where pa < N p / [ N - (rn - a)p]. Thus, since { u , } is weakly convergent, 11 u,() m , is uniformly bounded, and hypothesis (c) can be considered by (5.2.12) purely as a growth restriction on A , ( x , u, . . . , Dmu) for la1 < m - 1. To interpret hypothesis (b) the following result is useful.

,

(5.2.13) Let un+ u weakly in Wm,p(Q),and suppose that the functions A , ( x , y , z) defined on SZ X R" X [IRq satisfy the Caratheodory restrictions and the ellipticity condition

E

( ~ , ( x , y , r ) - ~ , ( x , y , z ' ) } ( z , - z> ~O }

(a.e.inQ),

14-m

for z #

c

2'.

Then if

J"(A.(X,

U",

. . . ,DWU,)

- A , ( x , u,, . . . , Dm-Iu,, D"u)){D"un - D O U )

+o

14-m

a s n - + w , D"u,+D%(for l a l = r n ) i n m e a s u r e o n Q . The proof of this result is easily demonstrated by first proving that if u,, -+ u weakly in Lp, and f(y) > 0 for y > 0, thenjaf(u,) + laf(u) implies un+ u in measure on SZ.

Now combining (5.2.11) and (5.2.13), we note that for arbitrary CT(S2) and 1aJ< rn - 1, if u,+u weakly in kbm,,(Q), then

'p E

A , ( x , u,, . . . , Dmun)Da'p+A,(x, u , . . I , D"u)D"cp

in measure on W. Thus since the functions A , satisfy the appropriate growth restrictions,

= ( R u , cp).

Consequently, Ru, + Ru weakly, and again the restriction on A implied by hypothesis (b) is a growth restriction on A , for 101 Q m - 1.

5.2C Removal of the coerciveness restriction Under certain circumstances, the result (5.2.3) can be substantially improved by replacing the coerciveness hypothesis ( f ( x ) , x)llxll- + w as llxll+ 00 with less restrictive conditions. Indeed, an operatorf E B ( X , X * ) satisfying Condition (G) may well map f onto a proper subset of X * , and thus fail to be surjective. The coerciveness hypothesis just mentioned was used in (5.2.3) to prove (a) the solvability of the Galerkin approximations (5.2.1),, and (b) an a priori estimate for the solutions of these approximate equations. Thus the improvement we now state will imply both (a) and (b) in some cases.

'

5.2

237

FINITE-DIMENSIONAL APPROXIMATIONS

(5.2.14) Theorem Suppose f is a bounded, continuous mapping of a real separable reflexive Banach space into itself, satisfying the following conditions: (I)

f is an odd mapping, i.e., f( - x ) = - f ( x ) for all x E X .

(11) Condition (G’) If x,+x weakly in X , f ( x , ) + y weakly in X * , and ( f ( x , ) , x,) + ( y , x), then x, -+ x strongly. Then, if I/f(x)ll 2 a for x E 3 2 , = (zl1lzll= R},.the equation f ( x ) = g has a solution in Z, for all g with I(gll < a .

Proof: Again the basic idea is to use the hypotheses of the theorem to ensure both the existence of the solutions of the Galerkin approximations and a priori bounds for the solutions so obtained. For then, since Condition (G’) implies Condition (G), our previous arguments imply that a subsequence of the solutions of the Galerkin approximations will converge to a solution of f(x) = g. To show that the Galerkin approximations (5.2.1), have a solution x, on Z, n X , for n sufficiently large, we first note that if g E X , 11 gll < a , and f ( x ) # rg for any t E [0, I ] and x E ax,, then for n sufficiently large, there are constants ,8 and N > 0 for which IlP,’(f(z) - tg)ll > p for t E [0, I ] and z E aZ, n X , for all n > N . Indeed, otherwise there would be sequences ( P ; ) , ( z , } , and { t , ) such that z, E Z, n X,, t , E[O, I], and (JP;[f(zk)- tkg]ll + O as k + 00. Thus, after passing to appropriate subsequences, we may suppose that z, + zo weakly, t, + to, and P , f ( z k ) + tog strongly in X * . Hence, for any w E X (P;f(z,), w) = ( f ( z k ) , Pflkw)+ ( t o g , w ) Also I(f(’k>

- ‘Oh’,

‘kW

-

w)l

IIf(’k)

-

rOgll

llPkw

-

wII

+’’

Consequently, expanding ( f ( z k ) - fog, P,w - w ) , we find f ( z k ) + f o g weakly in X * . Thus Condition (G’) implies that z, + zo strongly and so f(z,,) = tog. Consequently llzo(l = R , f(zo) = tog and IIf(zo)ll G I( gll, which contradicts the hypothesis that IIf(z)il > a > I( gll for z E 8 2 , n X,. This result shows that for n 2 N , first, the mappings P,*(f(z) - g) and P,’f(z) are homotopic on 8 2 , n X , , and secondly, with g = 0 that P,’f(z) # 0 on aZ, n X,. Thus, by (l.6.3), the Brouwer degree d(P,*f(z), 0, 2, n X , ) is an odd integer and hence not zero. Consequently, by the homotopy invariance of degree, d(P,*(f(z) - g), 0, Z, n X , ) # 0. This means that for n >, N ,P;(f(z) - g) = 0 has a solution 2, in 2 , n X , . Thus the Galerkin approximations (5.2.l), have a solution (z,} for n 2 N , and these solutions automatically satisfy the a priori bound JJz,jl < R. Thus the theorem is established. As in Section 5.2A, we now state hypotheses implying Condition (G’).

238


(5.2.15) Let f be a bounded continuous operator mapping X into X*. Then f satisfies Condition (G') if there is a completely continuous operator R:X+X*suchthatwithP=f- R (5.2.16)

for x, t E X , where f is weakly upper semicontinuous and satisfies f(0) = 0, and c ( r ) is real-valued, positive, and continuous, and c ( r ) + O if and only if r+O. ( P x - Pz, x - z ) + f ( x - z ) 2 c(llx - zll)

Proof: If x, + x, f ( x n ) +y, and ( f ( x , ) , x,) tation - shows that (Px, - P x , x, - x)+O. lim f(xn - x) Q 0. Thus (5.2.16) implies Since c is continuous and c( p ) = 0 if and in X.

+ ( y , x), then a short compuAlso, if x,+ x weakly, then that c(IIx, - xi]) = 0. only if /?= 0, x, + x strongly

For the. class of quasilinear elliptic operators discussed in (5.2.3), we prove the following condition analogous to Condition (G'). (5.2.17) Theorem Suppose A is a quasilinear operator satisfying the ellipticity condition (5.2.13), and in addition: (a) the associated abstract operator @ : ?@m,P(Q)+ W-,,, ,(a) satisfies the conditions mentioned in Section 5.2B together with the hypothesis (*) if u, + u weakly, then (P(u,, u,) - P(u, u,), u,) +0; (b) for fixedy there are integrable functions co(y)> 0 and c , ( y ) such that

z

Aa(X,Y9

z)za 2 co(y)lzIP

-

c,(y).

Ia(=m

Then the abstract operator 4" associated with A by duality satisfies Condition (G'). Proof: First we observe that since @ satisfies hypothesis (a), by the hypothesis of Condition (G'), @u, + @ u strongly in W P m,(a). , To show that u,+u strongly in @,,,P(Q), we shall prove that for la1 = rn, (i) the integrals JDJDmu,IP are equiabsolutely continuous and that (ii) Dau, + Dau in measure. The desired strong convergence then follows by Vitali's theorem. Now the result (ii) follows immediately from our assumptions, by virtue of (5.2.13). On the other hand, to prove (i) we use hypotheses (b) and (*) as follows: By hypothesis (5.2.5) and the fact that (gun,u , ) + ( @ u , u), we deduce (after a short computation) that (Pu,, u,) + ( P u , u). Then hypothesis (*) implies

(5.2.18)

(P(u, u,), u,)+(Pu, u).

Now by virtue of the definition and hypothesislb),

-


(5.2.18')

+

A , ( x , DYu,Dmu,,)Dau,, > co(DYu)IDmu,,IP.

cI(Dyu)

-

239

In(= m IYI< m

Using the facts concerning equiabsolutely continuous integrals, (5.2.13), the fact that Dau, Dau in measure, and the positivity of the expression A 4 ( x , y , z) z, implies that the functions &,,=, A , ( x , Dyu, Dmu,,)Dau, have equiabsolutely continuous integrals over Q . But in that case, the inequality (5.2.18') implies the same equiabsolutely continuous property for the integrals )Dau,,IP for la1 = m. Thus u , + u strongly in @ m , P ( Q ) and the result is proven.

z,,l=m

5.2D Rayleigh-Ritz approximations for gradient operators

The Galerkin approximations (5.2. l),, take a particularly elegant form if the mapping f is a gradient operator; i.e., f ( x ) = F'(x), where F ( x ) is a C real-valued functional defined on X . Indeed, in that case the solutions of (5.2.1),, are precisely the critical points of the functional g,, (x) = F ( x ) ( g , x) defined on X,,. Thus the powerful methods of finite-dimensional critical point theory discussed briefly in Section 1.6 become applicable in the study of the approximations (5.2. l),,. For historical reasons this approach is known as the Rayleigh-Ritz approximation. Here we shall illustrate the Rayleigh-Ritz method by reconsidering the class of nonlinear eigenvalue problems. Thus we study the solutions of the equation

'

(5.2.19)

h , & ' ( x ) =A,%

'(x),

x E {x

I

d ( x ) = const., x E X } ,

which can be obtained as limits of solutions of the Rayleigh-Ritz approximations

(5.2.20) h{"'P,*&'(x)= A?)P,*% ' ( x ) , x E { x 1 @ ( x )= const., x E X , } . First we prove a result concerning the approximations for the "first eigenvector" of (5.2.19). (5.2.21) Suppose that d (x) and % (x) are two C ' real-valued functionals defined on the separable reflexive Banach space X such that (i) % = { x I 8(x) = const.) is a bounded star-shaped set on X ; (ii) A x = U '(x) is a bounded continuous mapping that satisfies Condition (G'); (iii) B ( x ) = 3 '(x) IS completely continuous, with (21(x) = 0 if and only if x = 0. Then, c 1 = suprn % (x) is attained on % ' by an element T that satisfies

240

5


the equation (5.2.22)

A,AX

= A,BX,

where A, and A, are two real numbers not both zero. Furthermore, supmnxn 91 (x), and the elements (X,A,, A,) are limits of a subsequence (x,,, A?), A?)) satisfying (5.2.20) and the extremal property

c, =

sup %(x)

= %(x,,),

'7n n X,

x, EX,,

n 9R.

Proof: First we note that by hypothesis the sets % n X,, are compact, so that for each n, c,,,, = supqnxn ti3 (x) is attained by an element x, E (im n X,,. Consequently, for each n, there are constants A?), Ay) not both zero such that the triple (x,,, A?). Ap)) satisfies (5.2.20). Without loss of generality, we may suppose that lAf")l + lA$"l = 1, so that (after possibly passing to subsequences) A?) and A?) converge to A , and A, (say) with IAIl + IA,l = 1. Since X is a reflexive Banach space and the sequences { IIx,,ll), { IIAxJ} are uniformly bounded, we may suppose that, after again passing to suitable subsequences, x, + X weakly and Ax,, +y weakly. Assuming that A , # 0, we find that x , + X strongly since the operator A satisfies Condition (G'), y = ( A , / A , ) B x , and

Thus since % is closed, X belongs to " X ,and satisfies (5.2.22) by virtue of our discussion in Section 5.2.1. Finally, we rule out the possibility that A , = 0. If A , = 0, then '-21'(X) = 0, so that 2 = 0. But since 9I (x) is a weakly continuous functional, this implies that O = % ( x ) = n-m lim ?ih(x,,)= n+m lim sup % ( x ) = c,. "7ll n X ,

But this fact contradicts the hypothesis (iii) of (5.2.2 I). Consequently the desired result is proved. We now give brief mention of a result for more general critical points that can be obtained by finite-dimensional approximation and the Ljusternik-Schnirelmann category (see Section 6.6) (5.2.23) Theorem Suppose the functionals W ( x ) and $8 (x) satisfy the hypothesis of (5.2.21), and in addition suppose (a) W ( x ) and 3 ( x ) are even functions of x, and (x) are strictly positive for x # 0. (b) (@'(x),x) and 91 Then the real numbers (5.2.24)

cN = sup inf %(x) [AIN A


are critical points of

!It,

24 I

91 (x) restricted to

= {x

1

d ( x ) = const., x E

x),

where

(5.2.25)

[A],= { A I A

c 9R,

A

compact,

cat(A/Z,, 3R/Z2) 2 N } . Furthermore, for each fixed N there is a sequence of pairs (X,, , A,. ,,) with c,., = 33 (EN..)+ c, and X,.,, E 9 2 n X,, such that (X,,,, 1, A,,) satisfies the Rayleigh-Ritz approximations (5.2.l), as well as the minimax characterization ,c, ,= sup,An x n , inf, nx, $8(x). Furthermore, for each N , there is a subsequence of pairs (X,, ,, A,? ), converging strongly in X x R' to (EN,A,), where 7, E ti?-'(c,,) n '3R IS a critical point of 9~ (x) restricted to L3R and (X,, A,) satisfies the equation d '(EN) = A, 93 '(x,). For a proof of this result we refer the reader to the paper by Rabinowitz ( 1 973). 5.2E

Steady state solutions of the Navier-Stokes equations

The Navier-Stokes equations for three-dimensional steady flow of a viscous incompressible fluid in a bounded domain S2 in R3 can be written

A M + ( u . V ) U + Vp = g,

(5.2.26)

-

(5.2.27)

div u = 0,

(5.2.28)

ulan= P(x).

v

Here u ( x ) denotes the velocity vector of the fluid, v the viscosity of the fluid, p the pressure, g the external force acting on the fluid, and P ( x ) the value of u ( x ) on aS2. We demonstrate the existence of a solution for the system (5.2.26)-(5.2.28) by means of the Galerkin approximation scheme described in (5.2. I), for any value of v provided P ( x ) is suitably restricted. By a limiting process, one can show that the analogue of the system (5.2.26)-(5.2.28) for unbounded domains also has a solution. Actually we prove

(5.2.29) Theorem The system (5.2.26)-(5.2.28) possesses a generalized solution u ( x ) in the sense of Section 1.5, provided g E L2(0) and P ( x ) is the boundary value of a function P*(x) defined on with V P , Holder continuous, p* such that either (i) I V P*(x)l or I P * ( x ) l is sufficiently small, or (ii) &(x) = curl y(x), where y ( x ) E C ' ( G ) for a i l of class C2. The solution u ( x ) is smooth in S2 and at all sufficiently smooth portions of ail2, provided g is Holder continuous in

a

a.

Proot: We proceed by first representing the generalized solutions of (5.2.26)-(5.2.28) by means .of an operator equation of the form f(u) = g, where f isoa mapping on the Hilbert space H of solenoidal vectors w(x) whose components wi(x) E W , ,@). Then we show that

242


the operator f satisfies the hypotheses of (5.2.3) with p ( x ) suitably restricted. The regularity of the generalized solution then follows from the results mentioned in Section 1.5. First, suppose p ( x ) E 0. Then the weak solutions in H are in one-to-one correspondence with the solutions of the operator equation (5.2.30)

vu - 8 ( ~ U ) ,= j ,

where 4 (u, u ) and j are defined on H by the formulas

As demonstrated in Section 4.3, the operator 9( u ) = 9(u, u ) is a completely continuous mapping of H into itself. In addition, for u E CF(f2) n H ,

since div u = 0 and u (G), and by (5.2.32),

I

= 0. Thus for 0

< v < m, f,(u) = vu - 9(u. u ) satisfies Condition

(vu - Q u , u)/llull = YIIUII. so that f,(u) is cfercive. Consequently, by Theorem (5.2.3), the operator equation (5.2.30) has a solution ii E H; so that (5.2.26)-(5.2.28) has the generalized solution 6 in K. More generally, if p ( x ) # 0, suppose that, by virtue of the restrictions of the theorem, there is a constant c with y > c > 0 and (5.2.33)

I(s(P,,141,U)I

G CIIUII~

for u E H.

Then if we represent a generalized solution u of (5.2.26)-(5.2.28) by u = w w satisfies the equation (5.2.34)

YW -

+ 8. with w E 8,

+ %(w, P,) + a( B,, w)) =f,,

( Q(w, W)

where f, = f - /?, - 9( p,, p,). Now we show that Theorem (5.2.3) can be applied to (5.2.34). To this end, observe that each operator 8 (w, w), 4 (w, p,), and 4 (b,, w) is completely continuous. Thus for 0 < v < 00, the operator fJw) on the left-hand side of (5.2.34) satisfies Condition (G). Furthermore, by arguing as in (5.2.32), (8(w, p,), w) = 0, SO that by (5.2.33) (f"(W)I

W)/llWllA = vlIwII + (9(8,>w), w)/llwll >(v - c)llwll.

Consequently, f,(w) satisfies the coerciveness condition of (5.2.3), and therefore the equation (5.2.34) is solvable. Thus the system (5.2.26)-(5.2.28) has a generalized solution in H. Finally, we demonstrate (5.2.33), assuming that p ( x ) satisfies one of the restrictions of the theorem. First suppose that either Ip,(x)l < Mo or I V /3*(x)l < M I for x € 0 . Then by (5.2.31), using the fact that 1 p,(x)l < Mo,and Sobolev's inequality

I(4(P,,

< ~ollwllo,21lvwllo,2< ~ O C l l l W 1 1 % ~ and using the fact that 1 V p.(x)l < MI, w). w)l

I(4(P*, W)>

w)l

< M,llwI12,,

6 M,c:llwll%.

Thus (5.2.33) is satisfied if either v > Moc, or v > Mot:. On the other hand, if p.(x) = curl y ( x ) as in hypothesis (ii) of the theorem, we employ the inequality (5.2.35)

I V uI2

for u E k,,2 ( D )

with p = dist(xl

an)

5.3 HOMOTOPY, T H E DEGREE OF MAPPINGS. AND ITS GENERALIZATIONS

243

and construct a function h(r) E Cm[O, 0 0 ) depending on two parameters k , a such that (i) h(r) = 1 for 0 < t < ka and h ( t ) = 0 for t > (1 - k ) a , and (ii) as k+O, th’(t)+O uniformly with a and t. Then letting p ( x ) = dist(x, aQ), p.. = curl(h(p)y), p.. = h curl y - y x h ’ ( p ) V p. Thus p.. = p. on aQ, p.. = 0 outside a small neighborhood of aQ; and since p E C 2 , 8.. E C’.P(H). Furthermore, for any z > 0, Ip,!3..(x)l < z. By (5.2.35),

l(%(D**>u ) , u)l G U / P > u)l G ‘IlU/Pll0,211 Vullo.2 < ~ C , I I W 1 l 2 for all u E CT(Q).Thus for c sufficiently small, p.. satisfies (5.2.33). This completes the proof l%(PB.*3

of the theorem.

5.3 Homotopy, the Degree of Mappings, and Its Generailzatlons

5.3A Heuristics

Many problems involving a given nonlinear operatorf E C ( X ,Y ) can be studied by focusing attention on the topological properties of the mapping defined by f. In particular, replacing f with a simpler mapping f in the same (appropriately defined) homotopy class as f, sometimes makes it possible to solve the problem forf by solving an analogous problem for the simpler mapping f. This procedure is well known for mappings between finite-dimensional spaces. Thus, for a given analytic functionf(z) defined on a closed disk Z, = {zIzI < R } ; the number of zeros of f inside Z, is a topological invariant, provided of course that f(z) # 0 on = ( z 1 1zJ= R } . In fact by RouchC’s theorem, two analytic functions f and f + g define on Z, have the same number of zeros inside 2 , iff and f + g are homotopic on a Z, in the sense that If(z) tg(z)1 # 0 for z E a Z, and t E [0, I]. More generally, by the theorem of H. Hopf mentioned in Section 1.6.7, two continuous mappings f and f g defined on the ball 2 , = { x 1 1x1 < R } in R”’ withf # 0 on ax, = { x I 1x1 = R } have the same “algebraic” number of zeros in 2, if and only if they are homotopic. Here we shall discuss this homotopy approach to nonlinear problems for operators defined on bounded domains of Banach spaces. Immediately however the following formidable obstacle arises in the infinitedimensional case, unless we refine our notion of homotopy.

ax,

+ +

(5.3.1) Let H be a separable Hilbert space of infinite dimension. Let f and g be any two continuous mappings of the sphere 32, = {x 1 ((xIJH = 1) into itself. Then f and g are homotopic, i.e., there is a continuous mapping h ( x , t ) : ax, X [0, l ] + a X , such that h ( x , 0) - f ( x ) and h ( x , I)-g(x).

244


Proof: The basic idea is to construct a fixed point free mapping u of the ball Z, = { x 1 (Ix~I,, < l } into itself and thence a retraction r ( x ) of Z, onto ax,. Then the desired homotopy can be explicitly written in the form M x , 0 = r ( t g ( x ) + (1 - t)f(x)). To construct the fixed point free continuous mapping u, let (el, e2, . . . ) denote a complete orthonormal basis for H so that an arbitrary element x of H can be written x = x z r x i e i = (x,, x2, . . . ), with l(x(I2= ~ ~ , x , We ? ,

)

can then define u(x) = (dl - llx(12, x,, x2, . . . for x E 2 , . A simple computation shows that Ilu(x)I12 = 1; so that if u possesses a fixed point y E 2 , , I(yI(= 1. Consequently, if y = ( y l , y z ,. . . ), then yi = y i + , for i = 1, 2, . . . , while y , = 0. Thus y = 0, which contradicts the fact that llyll # 0. Hence u is a continuous mapping of Z, into 2 , that has no fixed point. To construct the retraction r ( x ) of 2 , onto ax,, we proceed as follows. For x E Z,, the line L ( x )joining x and u(x) does not degenerate to a point since u is fixed point free. Thus the line L ( x ) can be extended so as to intersect ax, in a point r ( x ) on the opposite side of u(x) from x but distinct from u(x). The mapping x += r ( x ) is the desired retraction because it clearly maps Z, onto ax, continuously and by construction leaves aZ, fixed pointwise.

5.38 Compact perturbations of a continuous mapping

Because of (5.3. l), we shall in Sections 5.3B-5.3D restrict attention to the following special class of homotopic deformations.

(5.3.2) Definition Let S be a closed subset of a Banach space X . Suppose f is a fixed continuous mapping of X -+ Y (a Banach space). Then go and g , are compactly homotopic on S (relative tof) if there is a continuous compact mapping h ( x , t ) : S x [0, 11-3 Y with go(x) = f ( x ) h ( x , 0) and g,(x) = f ( x ) h ( x , l), and such that g(x, t ) = f ( x ) + h ( x , t ) # 0 on S x [O, 11.

+

+

Clearly compact homotopy defines an equivalence relation on the class e , ( S , Y ) of compact perturbations of f E ef(S,Y ) = { g I g = f + K , K compact, g E C ( S , Y ) } . In the sequel we shall attempt to represent the resulting equivalence classes by computable topological invariants and to interpret these invariants in terms of the mapping properties of a given compact perturbation of the mapping!. In order to obtain a first result in this direction, suppose S is a closed subset of X and 0 denotes a component of X - S .

5.3 HOMOTOPY, THE DEGREE OF MAPPINGS, A N D ITS GENERALIZATIONS

245

(I) Notation and definitions Let k’,(S, Y ) = { g I g = f + K } , C?;(S, Y ) = { g ( g ECf (s,Y ) , g P O o n S } . Let 0 be a component of X - S. Then g E C;(S, Y ) is called inessentid (w.r.t. 0) if g has an extension E C;(O u S, Y ) . Otherwise, g is called essential. Moreover, we let C f (S, Y ) = { g I g = f + C, g # p on S } . Thus g is essential (with respect to 0) if every extension g E P,(O u S, Y ) has a zero in 0. Clearly, then to prove that a given g E C,(O u S, Y ) has a zero in 0, we need only show that g is essential (with respect to 0). By virtue of the next result we shall see that the same conclusion holds if some mapping j j , compactly homotopic to g (on S), is essential. Theorem The properties essential and nonessential (relative to 0) are invariant under compact homotopy.

(5.3.3)

Proof: It suffices to prove the result for an inessential mapping g E e;(S, Y ) . In that case, suppose g, 2 E-C;(S, Y ) are compactly homotopic on S and g admits an- extension G E C,(O u S, Y ) . Then we -shall construct an extension G E 2,(0 u S. Y ) of j j such that G and G are compactly homotopic on 0 u S. Since g and are compactly homotopic, there is a continuous compact mapping h ( x , t ) : S X I + Y satisfying the definition. Let To = S X [0, 11 u (S u 0 X {O}) and define h*: To+ Y by setting

h*(x, t ) =

u

G(x) - f ( x )

for x E S

h(x, t )

for x E S, t

0, t = O E [0,

11.

Then h* is compact and continuous on To, so that by the extension property (2.4.4) of compact operators, h* can be extended to a continuous compact mapping H * ( x , t ) of S u 0 X [0, 1]+ Y . In order to define the desired extension G, we must ensure that 6 # 0 on S u 0. Hence we let S, = { x I x E S u O , f ( x )= - H * ( x , t ) for t E [0, 11). Now S and S , are disjoint closed sets and by Tietze’s theorem there is a continuous function Y ( x ) : S u O+[O, 11 that vanishes on S , and takes the value 1 on S. Let H ( x , t ) = H * ( x , Y ( x ) t ) on S u 0 X [0, 11, and set G(x) = f ( x ) + H ( x , 1). Now we note that (i) G is an extension of g since if t = 1 and x E S

-

H ( x , 1) = H * ( x , Y ( x ) )= H * ( x , 1) = h ( x , 1).

-+

Thus 6 ( x ) = f ( x ) h ( x , 1) = g. (ii) G(x) # 0 for x E S u 0 and, in fact, more generally G(x, t ) = f ( x ) + H ( x , t ) # 0 for t E [0, 11, x E S u 0. Indeed otherwise x E s,,and thus

f ( x ) + H ( x , 0 ) =f(x)

+ H * ( x , 0 ) = G(x) = 0.

246


But this last equality is impossible since G(x) has no zero on S u 0. Thus not only does th_e extension G exist, but moreover since f ( x ) H ( x , 0 ) = G(x), G and G are compactly homotopic on S u 0.

+

To proceed further with this notion we note the following fact from finite-dimensional homotopy theory that is important in understanding our subsequent developments. (5.3.4) A continuous mappingf: S" + S m is essential relative to the open ball Z, = { x I x E R"+', 1x1 < l } if and only if the homotopy class [f E r,,(Srn)(of Section 1.6) is nontrivial. Proof: Suppose f is not essential, so that there is an extension F off to -

Z I with F ( x ) # 0. Then setting H ( x , t ) = F(tx)/lF(tx)l for t E [0, I], we note that by this homotopyf(x) is homotopic to the point H ( x , 0) = F(0) /lF(O)l, so that [f] E n,(Srn) = 0. Conversely, if [f] = 0, there is a homotopy h ( x , t ) of f ( x ) for t E [0, I ] with Ih(x, t)l = 1. Consequently F ( t x ) = h ( x , t ) is the desired nonzero extension off to I.

To proceed further it will be necessary to establish infinite-dimensional analogues of (5.3.4) yielding criteria for a given mapping g E C?,(S, Y ) to be essential. This task and more will be taken up in the next subsections for special choices off. First we shall choose f to be the identity mapping of a Banach space X onto itself; and in the second case, we suppose that f is a linear Fredholm mapping L of nonnegative indexp mapping X into Y . Indeed, example (5.3.1) shows that the infinite-dimensional result analogous to (5.3.4) is more subtle, and in fact we shall proceed in the sequel by considering only compact homotopies of a fixed linear operator. In such cases we shall, in fact, prove a generalization of (5.3.4) to the infinite-dimensional cases. But these generalizations will require, in general (for index L > 0), the use of the notion of stable homotopy class of the associated infinite-dimensional mapping. (This use of the word stable refers to the fact (1.6.8) that the homotopy groups { T " + ~ ( S " )n, = 1, 2, 3 . . . } are isomorphic for p > 0 only for sufficiently large n.) Thus, in general, to establish the essentialness of a given mapping in our class it does not suffice to examine on& the homotopy class of a sufficiently close finitedimensional approximation. (See Section 5.3D below.) 5.3C Compact perturbations of the identity and the Leray-Schauder degree

Let I denote the identity mapping of a Banach space into itself, D denote a bounded domain of X , and aD be the boundary of D.Then, as in

5.3 HOMOTOPY, T H E DEGREE OF MAPPINGS, AND ITS GENERALIZATIONS

247

the finite-dimensional case, it is possible to establish a one-to-one correspondence between the compact homotopy classes of e , ( a D , X ) and the integers Z by means of a function called the Leray-Schauder degree. In addition we shall establish a necessary and sufficient condition that a mapping g E C?, ( a D , X ) be essential; namely, that its Leray-Schauder degree be different from zero. The Leray-Schauder degree of a compact perturbation of the identity g = I C relative to a point p E X and D (written d ( l + C , p , D))can be defined by an analogue of the Galerkin approximation procedures of Section 5.2, assuming g(x) # p on aD. Assuming as known the facts (1.6.3) concerning the Brouwer degree dB for continuous mappings between spaces of the same finite dimension, we can define the integervalued function d(Z C , p , D ) in two steps:

+

+

Step 1: If the compact mapping C: D + X has finite-dimensional range (i.e., C ( D ) c X,, a finite-dimensional linear subspace of X ) , then assuming p E X,, we define the Leray-Schauder degree of I C at p relative to D by (5.3.5) d ( l + C , p , D ) = d,(Z + C , p , D n X,,)

+

for

I

+ C E PI( do, X ) .

In this case, the Brouwer degree dB is a finite integer since ( I + C)x # p for x E d(D n X , ) = a l l n X , , by hypothesis. Consequently, in this case, the Leray-Schauder degree is well defined provided the definition is independent of both the finite-dimensional subspace containing p and of C(D). Step 2: For a general compact mapping C: D + X , using (2.4.2), we approximate C by a sequence of compact mappings with finitedimensional range C,: D + X , ( X , a finite-dimensional subspace of X ) such that supxEDllC,x - Cxll < l / n . Then again, assuming p E X,, and the Leray-Schauder degrees of the approximations (as defined in step 1) exist, we define d ( l C , p , D ) = fl+W lim d ( l + C,,,p, D). (5.3.6)

+

Clearly the function d ( l + C , p , D )is well defined if the limit (5.3.6) exists and is independent of the approximating sequence C,. In order to justify the definition of d ( l + C , p , D )just given, we now provide the following

(5.3.7) Justification of step 1: We show that the integer d ( l + C , p , D ) defined by (5.3.5) is independent of both the finite-dimensional linear subspace X , containing p and C ( D ) . Thus, suppose the finitedimensional subspaces X , and X p both contain { p } u C ( D ) . Since

248

5


X p n X,, is also a finite-dimensional linear subspace containing p u C ( D ) , it suffices to assume that (i) the domain of I C is restricted to D n X,, (ii) X , c X,, and (iii) to prove that

+

dB(1 + C, p , D n X , )

=

dB(I + C, p , D n X , )

(a statement involving only properties of the finite-dimensional Brouwer degree). To this end, suppose dim X , = n, dim X p = n k , and a basis 3 in X p is chosen so that we may identify X,, with R " and X, with Rn+,. Since the Brouwer degree will be independent of the basis 9 chosen, (5.3.7) will be proven once we establish

+

Lemma Let A be a bounded domain in Rn+, and f be a continuous mapping of A +R". Then dB(I f,p , A) = dB(I f,p , A n R ") for all p E R", provided x + f ( x ) # p for x E aA. (5.3.8)

+

+

Proof: By virtue of the definition of the Brouwer degree, it suffices to suppose that f is a C ' mapping and that the Jacobian determinant of the mapping x + f ( x ) , J,,+,(x), does not vanish on the set u { x 1 x E A, x f ( x ) = p , for a fixedp E R"). Then by the properties of the Brouwer degree of Section 1.6

+

d B ( I + f , p ? A ) = E sgndetJ,+,(x)

(

U

= E sgndet J"o(x) =

2

l)

sgn det (J,(x))

0

= d(I

+f,p, An R"),

where Z, is the identity matrix in Rk and J,,(x) is the Jacobian determinant of the mapping x + f ( x ) on A n R". (5.3.9) Justiflcation of step 2: We begin by showing that for n sufficiently large, the integers d ( l + C,,, p , D ) (as specified in step 1) can be defined [i.e., ( I CJx # p for x E aD. Thus, we first find a number a > 0 such that infxE,, Il(I + C)x - pi1 > a , by virtue of the fact that I C E C?, ( a l l , X ) . Indeed, otherwise there would be a bounded sequence { x j } E a l l with (lxj Cxj - p(I +O. By the compactness of C, after possibly passing to a subsequence, we may suppose Cxj+y (say). Hence { x j } converges to z (say). Then, since aD is closed, z E aD and z + Cz = p , which is the desired contradiction. Now it is easy to show that I C,, - p E C?,(aD, X ) for n sufficiently large, since for x E a D and n no (say)

+

+

+

+

(Ix

+ C,x

- pi( > (Ix

+ cx - pi1 - 11 cx - C,Xll

> a - 4.2

fa.

5.3 HOMOTOPY. THE DEGREE OF MAPPINGS, AND ITS GENERALIZATIONS

249

Next we show that the numbers d,, = d ( I + C,,,p, D ) stabilize for n sufficiently large, by supposing that for any integers n, m > no, supaD11 Cx - C,,xll < + a and proving d,, = d, for any integer m > no. This fact will ensure that the limit (5.3.6) exists and is independent of the approximations C,,. To this end, suppose the integers n, rn > no, C,,(D) c X,,, C,(D) c X,, and X,,,, denotes the subspace generated by X , and X,,. Then by Lemma (5.3.Q (5.3.10)

d ( I + C,,.p, D n X,,) = d,(I

+ C,,,p, D n X,,,,),

and similarly for the mapping I + C,. Then, for x E D n X n + , , let h ( x , t ) + tC,x + (1 - t)C,,x - p so that on aD n X,,,,

=x

Ilh(x, t)ll

> IIX + cx -pi1 > a - + ( f a+ ( 1

- tllC,X -

- Cxl( - (1

-

t)llC,,x - Cx(l

t)a) > ; a .

Consequently, by virtue of the homotopy invariance of the Brouwer degree and (5.3.Q d,, = d,(l + C,,,p, D n X,,,) = d , ( I + Crn,p,D n X n + m ) = d m . We are now in a position to state and prove two major properties of the Leray-Schauder degree mentioned above. (5.3.11) Theorem Let f,g E cSy(aD, X ) , where D is a convex domain of X. Then f and g are compactly homotopic if and only if d(f,0, D ) = 4 g , 0, D ) . (5.3.12) Theorem Letf E Cy(aD, X). Thenfis essential relative to D if and only if d(f,0, D ) # 0. Thus if d(f,0, D ) # 0 the equation f ( x ) = 0 has a solution in D. Proof of (5.3.11): By virtue of the definition of the Leray-Schauder degree just given, iff, g E C,( d o , X ) and are compactly homotopic, then clearly the mappings f and g have the same Leray-Schauder degree at zero. Conversely, suppose f,g E C,( a D , X ) and d(f, 0, D ) = d( g , 0, D ) . Then by virtue of the definition just given, we may suppose that the ranges of the compact operators c = f - I and cI = g - I are contained in the same finite-dimensional linear subspaces X,,of X. In addition, by restricting the domains of f and g to X,,n D , we may also assume that d,(I + c , 0, X, n D ) = d,(I+ cl, 0, X,,n D ) . Then by the result of H. Hopf (1.6. 7), f and g are homotopic on X,,n aD; so that there is a continuous n aD]X function h ( x , t ) = x c(x, t ) defined on the closed set Z = [X,, [0, I ] such that h ( x , 0) = j ( x ) I x n n a Dand h ( x , 1) = g(x)lxnnaD.Choosing a basis (el, . . . , en) in X,,we can write c(x, t ) as an n-tuple of real-valued continuous functions c,(x, t ) ( i = 1, . . . , n). By Tietze’s theorem, each of these continuous functions can be continuously extended to a function C, on aD x [0, 11 with sup, (C,(x, t)l

+

250


+

~, , c~I ( x ~ ,t )[. ~We, then consider the function H ( x , t ) = x t)e,. H ( x , t ) - x is compact on aD x [0, 11 since it has a closed, bounded, finite-dimensional range, and in addition, for x E aD,H ( x , 0) = f, H ( x , 1) = g. Furthermore, H ( x , t ) is a compact homotopy between f,g in the class C?, ( a D , X ) since for t E [0, I ] and x E Z, H ( x , t ) # 0 by H ( x , t ) # 0 because x and C(x, t) are definition, while for x E aD n X,,, linearly independent. Consequently, f and g are compactly homotopic on aD.

-S

U

~

x:=,C,(x,

e,

Proof of (5.3.12): First suppose f E ( a D , X ) and d(f, 0, D ) # 0. Then by (5.3.1 l), for any compact perturbation of the identity g = I C defined on 5 and coinciding with f on aD, d(g, 0, D ) = d(f, 0, D ) # 0. Indsed, the convex combination off and g defines a compact homotopy of f into g. Thus it suffices to prove that the equation g(x) = 0 has a solution. By the definition of the Leray-Schauder degree, we may suppose that there is a sequence of mappings g,, = I C,,, where for each n, C,, is a compact mapping with finite-dimensional range and supD 11 C,,x - (2x11 < I/n. Furthermore, d(I C,,, 0, D )= d ( I C, 0, D ) # 0 for n sufficiently large. In addition, by restricting I + C,, to the intersection of D with a finite-dimensional linear subspace X,, of X , we find

+

+

+

d,(I

+ C,,, 0, D

+

n X,,) = d ( I +

C, 0, D ) # 0.

+

It follows then, by the properties of the Brouwer degree, that ( I C,)x = 0 has a solution x, E D n X,,. Next we show that ( x n } has a convergent subsequence with limit X,and g(Z) = 0. Indeed, {x,,} is bounded and for an appropriate subsequence ( x ? } , (Cx } converges. Thus "I (5.3-13)

llx,, + Cx,I

IICx, - Cfllx,,II + IIx,, + Cfl,x,,II

< Vn,;

so that as n + 00, {x,,} converges to K (say). Hence (5.3.13) implies that g ( 2 ) = 0. To prove the converse, we suppose that f E C?, ( a D , X ) , d(f,0, D ) =, 0, and f is essential: Then by (5.3.3) and (5.3.11), all mappings f E ( a D , X ) with d(f, 0, D )= 0 must be essential. Therefore, by the definition of Leray-Schauder degree, all continuous mappings f E C?, ( a D , X ) defined on D n X with f # 0 on aD n X must be essential. Hence the constant mapping is essential (a contradiction). Hence, in order for f E C?, (aD, X ) to be essential, d(f, 0, D ) # 0.

e,

Properties of the Leray-Schauder Degree We now describe the basic properties of the Leray-Schauder degree d( f, p , D),when considered as a function of the three variables f, p , and D. Then we use these

5.3 HOMOTOPY, THE DEGREE OF MAPPINGS, AND ITS GENERALlZATlONS

251

properties to discuss the computation of the degree for general classes of mappings in (2, ( a D , X ) . To begin we prove:

(5.3.14) Theorem Suppose D is a bounded domain contained in a Banach space X , a n d f - p E @:(all, X ) . Then the Leray-Schauder degree d(f,p , 0 ) is an integer with the following properties: (i) (Homotopy invariance) E 10, I ] is a compact homotopy = d ( h ( x , t ) , p , D ) for t E 10, 11.

If ( h ( x , r ) - p ) E ey (aD,X ) for 1 with h ( x , 0) =f, then d(f,p , 0 )

(ii) If p and p’ are in the same component of X

-

f ( a D ) , then

d(f,p , D )= d ( f , p ’ , D). (iii) d ( f , p , D) is uniquely determined by its values on aD. (iv) (Continuity) d(f,p , D ) is a continuous (locally constant) function off E C ( 5 ) (with respect to uniform convergence) and p E X . (v) (Domain decomposition) If D is the union of a finite number of open disjoint sets D, ( i = 1, . . . , N ) with aD,c aD and f ( x ) # p on U

N

aDj, then N

(5.3.15)

d ( f ,p , D )=

2 d(f, p , 0,). i= 1

(vi) (Excision) If A is a closed subset of 5 on which f ( x ) # p , then d ( f , p , 0 )= d ( f , p , D - A). (vii) (Cartesian product formula) If X = X , 63 X , with Dic X i , f = ( f l , f,) with fi: Dj Xi (i = 1, 2), D = D ,X D, and p = ( p , ,p,), then d(f,p , D )= d(f,,p , , D,)d(f,, p2, DJ,provided the right-hand side is defined. (viii) (Index theorem) If the solutions of f ( x ) = p are isolated in D , then d(f,p , D )= x j d ( f , p ,O;), where 0; is any sufficiently small open neighborhood containing only one solution, and all solutions are contained in U ;Oj.

-

Proof: (i): This result is a restatement of (5.3.1 1). (ii): First we observe that since f is proper on all, f ( a D ) is closed, and consequently each of the components of X - f ( a D ) are open, arcwise connected sets. Let any such component be denoted D,,then there is an arc p(r) in D j for t E [0, I ] joining p and p’ and avoiding f(aD). Thus, by (i), for t E 10, 11, d(f, p , D )= d ( f , p ( t ) ,D )= d ( f , p ’ , D). (iii): Let fo be a compact perturbation of the identity that agrees with f on aD. Then by (i), h ( x , t ) = ‘f+ (1 - t)fo is a compact homotopy joiningf p , 0) = d(f,p , 0). and fo, 4fo, (iv): This is immediate from the definition and (i). (v)-(vii): Since each of these facts is valid for the Brouwer degree, the

252


definition of d ( f , p , D ) implies the validity of each for mappings f with finite-dimensional range, and so, by approximation, for allf E e y ( a D , X ) . (viii): The index theorem follows immediately from the excision and domain decomposition properties of degree using the fact that if the solutions of f ( x ) = p are isolated, they are finite in number. We now apply the eight properties of the Leray-Schauder degree just established to investigate classes of mappings that have nonzero LeraySchauder degree relative to a bounded domain D of a Banach space X . By virtue of (5.3.12), such mappings will be essential relative to D ,and hence of particular importance. (5.3.16) Theorem Supposef E e y ( a D , X ) . Then (i) If D contains the origin and f = I - C is a linear homeowhere ,f3 is the number of eigenvalues morphism, then d( f,0, 0)= ( of C (counted according to multiplicity) in the interval (1, co). More generally, assumingf is a compact perturbation of the identity and i f f is a homeomorphism of D onto itself, then d(f, 0, D )= -+ 1 provided D contains the origin. (ii) I f f is asymptotically linear, i.e., there is a compact linear mapping C with Ilf(x) - x Cx]l/]lxll+ O as llxll+ 00, and D is a sufficiently large domain containing the origin, then d(f, 0, 0 ) = (- 1)@, where /3 is the number of eigenvalues of C (counted with multiplicity in the interval (1, co)), provided I - C is a linear homeomorphism. (iii) Iff is an odd mapping and D is a symmetric domain containing the origin, then d(f, 0, D ) is an odd integer. More generally, if the parity assumption is weakened to f ( x ) # tf( - x) for t E [0, I ] and x E aD, then d ( f , 0, 0 ) is an odd integer. (iv) Suppose f = I + N , with N compact, and that all solutions of the family of equations f t ( x ) = x t N x = 0 for i E [0, 11 lie in some fixed bounded domain D that contains the origin, then d(f,0, D )= 1. (v) If X is a complex Banach space and f is complex analytic, then:

+

+

(a) d ( f , 0, 0 ) 2 0 ; (b) a necessary and sufficient condition for d(f, 0, D )> 0 is that 0 E ~ ( D ) and ; (c) a necessary and sufficient condition for d(f, 0, D ) 2 2 is that either the equation f ( x ) = 0 has more than one solution in D or at the unique solution xo of f(x) = 0 the linear operator f ' ( x o ) is not invertible. Proof: (i): Suppose f = I - C is a linear homeomorphism with f E (2, (aD,X ) . Let X , be the direct sum of the invariant subspaces of C corresponding to the eigenvalues in (1, a).Then, by hypothesis, dim X ,

5.3

HOMOTOPY. THF. DEGREE OF MAPPINGS. A N D ITS GENERALlZATlONS

253

= p < 00. Hence X = X i CB X,, where X , is invariant under f, and by (5.3.14), i f f i =fix, ( i = 1, 2) a n d f = (fl,f2), then

(5.3.17)

d(f, 0, D ) = d(f,, 0 , D

n X,)d(f,.

0 , D n X,).

Now on the finite-dimensional space D n X i , f l is (compactly) homotopic to - I . Indeed, setting h ( x , t ) = -(1 - r)x + [ ( I - C)x = (2t - 1)I- rC for f E [0, 11, we observe that h ( x , t ) # 0 on aD n X i this is clear for t = 0 and follows by hypothesis for t = 1 and for other t values since otherwise C would have an eigenvalue on the interval ( - co, 1) on Xi. Thus by (1.6.3), d(f,0, D n XI) = (- 1) fi. Next, we observe that f2 is homotopic to the identity I on aD n X , by virtue of the compact homotopy g(x, t ) = x tCx. Clearly, g(x, t ) # 0 on ( a l l n X,) X [0, 11 since if it did vanish there, C would have an eigenvalue in X , on the interval ( I , 00). Consequently, d(f2,0, D n D2) = 1; and finally by (5.3.17), d ( f , 0, D ) = ( Next, suppose f is a homeomorphism with inverse f - ' and f ( D ) = D . Then, by definition, since D contains the origin, d ( l , 0, D ) = 1, 1

=

d(ff-'. 0, D )= d(f,0, D ) d ( f - ' , 0, D )

so that d(f,0, D ) = ? I . (ii): By the homotopy invariance of degree on a D , the boundary of a sufficiently large domain containing the origin, f is compactly homotopic to I - C. Indeed, if f = I - C - N is linearly asymptotic to I - C, then IINxll/llxll + O as IIx/l+ 00. Therefore, the compact homotopy h ( x , r ) = x - { Cx + t N x ) joining f and 1 - C does not vanish on aD. Indeed, h ( x , t ) = 0 implies I/xo- Cxoll = toll Nxoll for some to E [0, I ] and xo E a D ; and since I - C is invertible, for some constant p > 0, pllxoll < llxo Cx,l( = toll Nxoll < 11 Nx,ll. This last inequality contradicts the fact that IINxIl/llxlj+O as lixll+00, since /3 is independent of x. Thus by the homotopy invariance of degree and (i) above, d(f,0, D ) = d ( l - C , 0, D ) = ( - 1)P.

(iii): Supposef= I + N is an odd mapping with N compact. Then if N had finite-dimensional range, d(f, 0, D ) would be an odd integer since by virtue of (1.6.3), the Brouwer degree has this property. Thus in the general case it suffices to prove that N can be closely approximated by odd compact mappings with finite-dimensional range N,. In fact, we shall construct such approximations with IINx - N,xll < c for all x E D and any E > 0. To this end, suppose M is a compact finite-dimensional approximation to N on D with' IjMx - Nxll < E . Then the operator N , x = 4 { M ( x ) - M ( - x)} is clearly an odd compact mapping with finitedimensional range and llNx - N , ~ l lQ

4 (IIMx - Nxll + I I M ( - x )

- N ( - - ~ ) l l )Q E.

254


To prove the more general result, we use the homotopy invariance of degree, by showing that the mapping f = 1 N ’ (say) is compactly homo{ N ’ ( x ) - N ’ ( - x ) ) . Intopic on ao to f l ( x ) = f { f ( x ) - f( - x ) } = x deed, consider the compact homotopy defined on aD

+

++

h ( x , t ) = (1 + t ) - I { f ( x ) - tf(-x)} =x+(1 +t)-]{N’(x)- tN’(-x)}.

The fact that f ( x ) # Af(- x ) for t E [0, 11 and x E aD implies h ( x , t ) # 0 for t E [0, 11 and x E aD. Consequently, by the result of the above paragraph, d(f, 0, D )= d ( h ( x , t ) , 0, D )= d ( t = odd

( f ( 4- f ( - X ) ) ,

0,D )

integer.

+

(iv): The compact homotopy fi = Z t N joins f and I , and by hypothesis does not vanish on aD X [0, 11. Therefore, by the homotopy invariance of degree,

d(f,0,D )= d(fi, 0, D ) = d(Z, 0, 0 ) = 1. (v): To establish (a) and (b) for a general complex analytic operator f E (aD, X ) , we first apply the result (1.6.2) to conclude that the set (I = { x I x E D , f ( x ) = 0 } is finite. Thus, denoting the points in u as

e,

XI,

.

,x, n

(5.3.18)

d(f,0, D ) =

d(f,0, O i ) , i= 1

where the Oi are small (pairwise disjoint) open neighborhoods of D such that xi E Oi for i = 1, . . . , n. Thus to prove (a) and (b), we need only show that each term on the right-hand side of (5.3.18) is positive. To this end, observe that by adding an arbitrarily small complex linear map L i ( x - x i ) of finite rank to f, we may suppose that f’(x,) is a linear homeomorphism. Then since f ( x ) = f’(xi)(x - x i ) + O(11x - xi1I2), the homotopy invariance of degree imply that if Oi is a sufficiently small neighborhood of xi,

d(f, 0, O i )= d ( f ’ ( X i ) , 0, O i ) = (- 1)O. But, since !’(xi) is a linear isomorphism defined on a complex Banach space X , 0 is even. Thus (a) and (b) are established. Finally, we observe that property (c) holds for f = I N , where N is a compact complex analytic operator with finite-dimensional range since it holds for the Brouwer degree (cf. (1.6.3(x))). Furthermore, by the argument used to prove (a) and (b), if the solutions of f ( x ) = 0 are not unique, then d(f,0, 0)Z 2. Therefore, it suffices to prove (c) under the assumption that

+

5.3 HOMOTOPY, THE DEGREE OF MAPPINGS, A N D ITS GENERALIZATIONS

255

the solution xo = 0 of f ( x ) = 0 exists and is unique in D. f’(0) is not invertible on X I , X,nKerf’(O) = 0 and X = Kerf’(0) @ X , . Then Y(0) = I + C with C compact, and we set C, = PC and C2 = ( I - P ) C , where P and I - P denote the canonical projections of X onto Kerf’(0) and X , , respectively. Then I + C, is invertible on X . Moreover, since C, is compact, there is a continuous complex valued function a ( t ) defined for t E [0, 11 with a(0) = 0 and a ( l ) = 1 such that the operators I - a ( t ) C 2are invertible. (Indeed C possesses discrete eigenvalues whose only possible limit point is zero.) Then by the analytic form of the inverse function theorem (3.1.1), if f ( x ) = x Cx + R x , where R x = O(llxl12), the operator I + a(t)[C,x + R x ] has a uniquely defined inverse h ( x , t ) = x p ( x , t ) joining h ( x , 0) = x and h ( x , 1) = [x + C2x + Rx1-l defined and continuous jointly in a neighborhood U of the origin for t E [0, 11. Furthermore, since x = h ( x , I ) + a ( t ) ( C z+ R ) ( h ( x , t ) ) , x - h ( x , t ) = p ( x , t ) is compact in x and f . Also, iff o h ( x , t ) = 0 in x [0, I], then h ( x , t ) = 0 so that x = 0. Thus, by virtue of the homotopy invariance of degree for t E [O, 11,

+

+

v

d(f,0, U ) = d(f h ( x , l ) , 0, U ) = d(f h ( x , l), 0, u ) 0

=

0

d ( I + C , h ( x , I), 0, U ) .

Now the mapping C , h ( x , I) is a compact complex analytic mapping with finite-dimensional range. In addition, the Frechet derivativ? of x + C , h ( x , 1) at x = 0 is 1 C , ( I + C2)-’. Thus the mapping f ( x ) = x + C , h ( x , 1) (restricted to the associated finite-dimensional subspace of X ) is a compact complex analytic pefturbation of the identity with finitedimensional range and such that f’(0) is not invertible, while d(f, 0, 0) is defined. Thus d(f,0, D ) > 2, by applying the analogous result (1.6.3) for the Brouwer degree. Finally we examine the conditions under which a mapping f E C?, (do, X ) is inessential, relative to D.

+

(5.3.19) Theorem Suppose f = I + C is a compact perturbation of the identity defined on a bounded domain of a Banach space X . Then, if defined, (i) d ( f , p , D ) = 0 if f maps D into a proper subspace X ’ of X and

p

EX’;

(ii) d(f,p , D ) = 0 if f ( x ) # p in D . The converse is true provided X is a complex Banach space, and f is complex analytic.

Proof: (i): Let A,, be the open component of X - f(dD) containing p . Since f(0)is contained in a proper subspace X ‘ of X , there is a point q E Ap that is not in the range off. Otherwise Ap c f ( D ) . Now by virtue of

256


(5.3.14(ii)) and the argument used in proving property (5.3.12) of the Leray-Schauder degree d ( f , p , D )= d(f,q, D ) = 0. (ii): This fact is an immediate consequence of the argument used in proving (5.3.12) and (5.3.16). 5.3D Compact perturbations of a linear Fredholm mapping and stable homotopy

Let L be a fixed bounded linear Fredholm operator of nonnegative' indexp mapping a Banach space X into itself. Let D = ( x 1 llxll < 1) and aD = ( x I(x(I= I}. Then we shall represent the compact homotopy (aD,X ) by means of known homotopy invariants. In addiclasses of tion, in certain cases, we shall determine special necessary and sufficient conditions that a mapping g E (3; ( a D , X ) be essential, and apply this to solvability questions involving operator equations. To accomplish these goals, notions concerning the sequence of homotopy groups of a sphere S", { T , , + ~ ( S " ) }(as n runs through the positive integers with p fixed) are of prime importance. As was mentioned ) isomorphic finite in Section 1.6, for fixed p > 0, the groups T " + ~ ( S " are Abelian groups provided n > p + 1, and this isomorphism is given by a canonical one, the so-called Freundenthal suspension E : vn+ p ( S " ) + v n + +, p ( S n + ' ) . These isomorphic groups are called the pth stable hornotopy groups of S". It is useful to find a simple analytic expression of the Freundenthal suspension Ef of a giyen representative f: S " + p + S" of [f] E r,+,,(S"). To accomplish this let f be any continuous extension of f to the interior of S" c ( W n + ' , and let Ef: S n + p + ' + S n + ' be the mapping

I

ei

(*) Ef(xl, . . . with k = n + p .

=

(r"

(

~

1

7

+

..

-%+I), % + 2 ) / I ( J

(XI,

...

9

xk+2))I

This is easily justified by virtue of the geometric definition of suspension given in Chapter 1 and the fact that the homotopy class of Ef depends only on [f]. Moreover the expression (*) can be easily extended to give a simple analytical expression of the various iterates of Freundenthal suspension operator Ekf ( k > 0). The reader should note that homotopy considerations are of little value in the case of Fredholm operators L of negative index. Indeed, in Section 3.1 we proved that C' operators of the form L + C must have a nowhere dense range. Thus the simple idea of perturbing a point p in the range of L + C to a nearby point p', without affecting the solvability of L + C breaks down in total agreement with (3.1.46).

5.3

HOMOTOPY. THE DEGREE OF MAPPINGS. A N D ITS GENERALIZATIONS

257

In case p = 0, 7i,,(S")=Z (the additive group of integers) and the resulting homotopy classes of mappingf: S" S" are well behaved under suspension, in the sense that the essentialness of such a mapping is preserved under suspension. This is no longer true, in general, i f p > 0, and indeed interesting homoiop,v properties of a mapping f : S n + p + S" may be lost upon iterated suspension. A case in point concerns the group 7i3(S2) which is known to be isomorphic to Z , while r4(S3)= Z,, thus if [ a ]is a generator of n3(S2),E [na]= 0 whenever n is even. Given a mapping j : S n + p + S " we can thus associate with it its homotopy class [fE rn+,(S"). Moreover, we consider its sequence of iterates by Freundenthal suspensions Ekf and associated homotopy classes E k [ f ]E T , , + , + ~ ( S ~ +Thus ~ ) . we callEk[f] the stable homotopy class o f f provided the integer k is so large that n + k > p + 1. We are now in a position to prove:

(5.3.20) Theorem (Svarc, 1964) Suppose L is a fixed linear Fredholm operator of nonnegative index p mapping X into Y . Then the compact homotopy classes !C (do, Y ) are in one-to-one correspondence with the elements of the pth stable homotopy group 7i,,+,(Sn) ( n > p + I). Proof: The basic idea is to repeat the arguments given in the construction of the Leray-Schauder degree, substituting the properties of stable C ( a l l , Y ) , we may homotopy for the Brouwer degree. Thus given f E : suppose that f = L + C , where (i) C has finite-dimensional range contained in the (finite-dimensional) subspace y,, of Y (provided n > p + I ) by (2.4.2), and (ii) by (1.3.38) L is surjective. Since L is a linear Fredholm operator of index p . we may write X = Ker L G3 X I , where dim Ker L = p and L : X I + Y is a bounded linear homeomorphism with inverse L - I . By we then restricting the domain of f to do,, n {Ker L @ L - ' ( _Y,,+,)}, Y,,, I defined by f ( x ) = f ( x o + xI) obtain a natural mapping f: aD,,+, = L x , + C(x, + x I ) . Furthermore, since f E 6?; ( a D , Y ) , not only does f ( x ) # 0 for x E aD, but also there is a positive number a > 0 with infXEaDIIf(x)ll 2 a > 0. Indeed, otherwise by arguing as in the proof of (5.3.9) and using the properties of linear Fredholm operators, there would be a sequence ( z,,} E aD with xn + K and llf(xn)ll +O so that f(2) = 0 and X E aD. Thus f ( x ) # 0 for x E do,,+,, and we can define a natural mapping of S"+P+ S"; f o ( x ) = f ( x ) / I l j ( x ) l l for x E S n + p . The correspondence 7 between the homotopy class o f f E eL(do, Y ) and 7in+,(Sn) is defined by setting ~ ( [ f ]=) [f,]. In order to show that 7 is well defined, it is necessary to prove that 7 is independent of the finite-dimensional subspace Y , defining [f,].To this end, suppose Y,,, and Y m + l are two subspaces with n, m > p + I , both containing C ( a D ) and let f, and g, denote the associated mappings between spheres. Then the subspace .--$

258


Y,,,, n Y,,,,, contains C ( a D ) , and both fo and go can be regarded as extensions by iterated suspension of the same mapping yo: aD

n {Ker L n K1( Y,,, n Y , + , > >

+

Y,+I ny,,,+I.

By a fundamental topological result, the homotopy classes [fO], [ g o ]of both fo and go depend only on [ y o ] . Hence, both [fO] and [go] are in the same

stable homotopy class, by virtue of the properties of the Freudenthal suspension operator. To prove that the correspondence T is one-to-one, suppose that the two mappings f,g E !?C (aD,Y ) are compact, finite-dimensional perturbations of L with [ ~ f=][ ~ g ] We . wish to prove that f and g are compactly homotopic in !?C (do,X ) . By arguing as in the first part of (5.3.4), we may suppose that the ranges of f,- L and g - L are contained in the same finite-dimension_al subspace ,Y of Y . Then there is a compact homotopy L x , C(x, t ) : X x [0, l ] + Y from f to g. By repeating the argument of the proof of (5.3.1 I), the Tietze extension theorem ensures that the mapping L x , C ( x , t ) can be extended to a compact homotopy in e;(aD, Y ) . It remains to show that if f, and f2 are compactly homotopic in (aD,Y ) , then ~ [ f ,=] ~ [ f , ]T. o this end we first suppose that f,= L C, and f2 = L + C, are such that each compact operator C, ( i = 1, 2) has finite-dimensional range. Then, as in (5.3.1 l), we observe that the compact homotopy h(x, t ) = L C ( x , t ) joiningf, andf, may be chosen such that C ( x , t ) has a fixed finite-dimensional range for all t . Then by the definition ] .[f2]. For the general of 7[fi] and 7[f2]given above, it follows that ~ [ f , = case, we first note that the compact homotgpy classes [f,] and [f2] each have representatives of the special form fi = L C, ( i = 1, 2) with C, compact. Then, the argument just given shows that the correspondence ~ [ f i=] ~ [ f i (]i = 1, 2) does not depend on the particular representative fi chosen.

+

+

+

e:

+

+

(5.3.20’) Corollary Under the hypothesis of Svarc’s theorem,f= L + C is inessential if and only if the natural mapping T constructed in the proof of (5.3.20) is such that ~ ( f=)0. Proof: Let g = L - co, where co is a constant map into Y , where co is ‘3 Ker L, chosen so that the image of the set { x , I x = x , xo, X = X , 3 x , E X I , (Ix(I < I } under L does not contain co. Consequently, by our construction of T in (5.3.20) T ( g ) = 0. Now suppose ~ [ f =] 0 so that ~ [ f=] T [ g ] . Then f and g have the same compact homotopy class. By (5.3.3) f is necessarily inessential since g is. On the other hand, if f is inessential, then necessarily f is compactly homotopic to g but T [ g ] = 0 and so ~ [ f=] 0. A simple yet interesting construction of a mapping g E ( a D , Y ) with

+

eL

5.3 HOMOTOPY. T H E DECREE OF MAPPINGS, A N D ITS GENERALIZATIONS

259

index L = p can be obtained as follows. Suppose 4 is compact and I + 4 is a mapping of a Banach space X onto a linear subspace Y of codimension p , while x + +(x) # 0 on the boundary of the unit sphere ax, of X . By virtue of (5.3.19), the Leray-Schauder degree d ( l + 4,0, ax,) = 0; and so the mapping I + 4 is inessential relative to 0. However, by restricting the range of I and 4 to Y and regarding g = I + 4 as a mapping from the unit sphere of X into Y , one can study the mapping properties of I 4 by means of g . Indeed, denoting by L and C the operators obtained by restricting the range of 1 and X to Y , g E C, (aD,Y ) ; and L can be regarded as a linear operator of index p . In fact, we shall determine necessary and sufficient conditions for such a mapping g to be essential.

+

Application to equations involving operators with singular points In order to utilize Svarc's theorem (5.3.20) we attempt to apply it to study the soloability of a simple class of semilinear operator equations. The class of operators we consider consists of uniform& bounded compact perturbations N of a fixed linear Fredholm operator L E L ( X , Y ) of index p . If p > 0, (5.3.20) does imply a solvability criteria for the equation Lu N u = 0 provided the associated stable homotopy class r of L N is nontrivial when 7 is regarded as an element of T~,~(S") for n sufficiently large. Such a result is clearly difficult to apply in case p > 0. Indeed if p = 0, as already mentioned, the essentialness of good finite-dimensional approximations to maps of the form L + C is preserved under iterated suspension. However this is no longer true, in general, if p > 0. Thus it is necessary to supplement the abstract theorem by using simplifying hypotheses that can often be verified in concrete problems. Thus in the sequel we shall assume not only the uniform boundedness of the operator N but also the following restriction on the asymptotic behavior of N .

+

+

Hypothesis (A): Let X = Ker L CB X I and Po be the canonical projection of Y onto coker L . Then IIP,N(x, + xl)li # 0 whenever x I E X I is uniformly bounded and so E Ker L is sufficiently large in norm.

This assumption is generally valid for those operator equations that admit a a priori estimaies. We are now in a position to prove the following improvement of (5.3.20).

(5.3.21) Theorem Let D, be a ball in a Banach space X of sufficiently large radius R and suppose L + N E (aD,, Y ) , where N is uniformly bounded on X and satisfies the above Hypothesis (A). Then L + N is essential if and only if the stable homotopy class of the mapping b(a) = p ( a ) / l p ( a ) l : S d - ' + S d * - l is nontrivial, where p ( a ) = P o N ( R a ) , d = dim Ker L , d, = dim Ker L*, and a is an element of norm I on Ker L .

260


Proof: We show first that an operator f = L + C satisfying the stated hypotheses can be deformed on aD, by a compact homotopy to the form f ( x o , x,) = ( L x , , P,Nx,): X I CB Ker L + Y , CB coker L and then apply the construction given in the proof of Svarc's result (5.3.20) to show in case Lx, = x,, that the homotopy class o f f regarded as a singularity free map of aD, corresponds to the stable homotopy class of the normalized map fi associated with p ( a ) = P , N ( R a ) . Then, the result will follow easily since the essentialness of a mapping f is invariant under linear homeomorphisms.

-

Step 1: Thus to demonstrate the compact homotopy joining f to f, we write f in the form f = ( P If, P , f ) , where P I and Pa are the canonical projections of Y onto Y , and coker L , respectively. Then we consider the

compact homotopy h(x, t ) = (Lx,+ t P , N x , PoN(xo + tx,))

joining f and j. Assuming the uniformly bounded mapping N satisfying Hypothesis ( A ) , h ( x , , t ) # 0 on aD, x [0, I] provided R is sufficiently large. Indeed if both P , N ( x , + t x , ) and L x , + t P , N x are zero, both ((xoIJ and llxlll (and consequently Ilxil) must be sufficiently small. Step 2: Now we observe that since L : X I - + Y , is a - l i n e a r homeomorphism, we may suppose, without loss of generality, that f ( x o , x,) = (x,, P,Nxo) and that coker L c Ker L , so that Ker L = coker L CB W . In this case we assert the homotopy class off, [f], corresponds by means of the construction in the proof of (5.3.20) to the stable homotopy class f of the normalized map fi associated with P , N ( R a ) when [ fi] is regarded as an element of rd-, ( S d * - ' ) . To verify this following the construction of the correspondence T in (5.3.20), we replace Lx = x, with the surjective map Lx = x, E O , where E > 0 is small and_ x = x , t? w with t? E coker L = V , w E, W and Cx = P , N ( x , ) with Cx = P o N ( x , ) - w. Thus Range C coker L , and we can regard C as a mapping of X I CB Ker L + X , CB coker L (with X , a linear subspace of dimension K so chosen that K n > p + 2). Setting S N + P=. { x [ (1x11= 1, x E Ker L CB L - ' ( V CB X,)), since L - ' ( V CB X,) = V CEl X,, we can identify S N + P as the unit sphere in X , CB V @ W . Consequently [f] coincides with the homotopy class of the normalization of the mapping

+

+ +

+

f (x) = ( x K

+ ED) + PONxO

- €0 =

x,q

+ P,NxO,

so ~ [ f =] the homotopy class of the Kth iterated Freudentahal suspension homomorphism of [ ,G(a)], E K [fi], and by our choice of K, E K [jl] is stable. Step 3: Finally to prove our result we note by (5.3.20') that

j

(and

5.3

HOMOTOPY. T H E DEGREE OF MAPPINGS, A N D ITS GENERALIZATIONS

261

consequently f ) is inessential if and only if the stable homotopy class of F(a) is zero. The utilization of unstable homotopy groups for operator equations The result (5.3.21) just obtained can be sharpened still further in analogy with (5.3.4) by attempting to eliminate the term “stable” from the statement of the homotopy criterion for solvability. Indeed in case the index of L , p = 0, (5.3.21) implies that the operator equation Lx N x = 0 is solvable if the homotopy class of the map G(a) is nontrivial. On the other hand, for p > 0, a short study of the tabulation of the homotopy groups T , , + ~ ( Sas ~ )can be found in Toda [1961] shows that much information is often lost in using the stable homotopy class of j l as a criterion for solvability. It turns out that the nontriviality of the homotopy class of ,Li does not ensure solvability, but rather the following sharp result holds.

+

(5.3.22) Theorem Suppose that L E L ( X , Y ) is a linear Fredholm operator of nonnegative index p and that N E C’(X, Y ) satisfying Hypothesis (A) as well as the property that )I N x l J and 11 N’(x)ll are uniformly bounded. Suppose for some E > 0 the following inequalities hold off some finite-dimensional space W = Ker L @ V (5.3.23) IILwll > ( c + ~ ) l l w l l , llPN’(u)wll Q C I I W I I , where P is canonical projection of Y onto L ( X / W ) . Then, if dim V = m , the equation L.w + N x = 0 is solvable provided the mth iterate of the Freudenthal suspension homomorphism of the homotopy class of jl, [ ,Li], defined in Section 5.3D above, E m [fi], is a nontrivial element if T d + m - I(’ d . + m - l ). In particular the equation is solvable provided [ p ] is nontrivial if eitherp = 0, m = 0 or more generally E m is an isomorphism of T ~ - , ( s ~ *into - ’ )T ~ + ,~( -s ~ +). ~ * - ’ Proof: The fundamental idea used here is the immediate application of the reduction lemma (5.1.9) to replace the solvability question with a finite-dimensional problem. The finite-dimensional problem is then resolved by means of the properties of the Freudenthal suspension mapping. To carry out this idea we write X = W @ W , and observe that the reduction lemma (5.1.9) implies that the solvability of Lu + N u = 0 can be reduced to a study of the equation

(5.3.23’)

Lw,

+ P,N(wo + w,[w,,]) = 0,

where P , is the canonical projection of Y = L( W , )CB Yo+ Yo. Moreover the reduction lemma and the uniform boundedness of N u over Y implies that Ilw,(w,)ll is also uniformly bounded over x . Now we study the system (5.3.23’) by once more decomposing it into two parts, one part on Ker L and the other part on V . Thus writing W = Ker L CB V and w o E W as

262


+

w o = xo u, the left-hand side of (5.3.23') can be written as the mapping d ( w ) = (Lu P , N ( w ) , P,N(w, wI(wo))). Next we observe that on a sphere of sufficiently large radius R in W , g ( w ) can be homotopically deformed to the mapping go(w) = (Lu, PoN(xo)) by the homotopy

+

+

h ( w , t ) = (Lu + t P J ( w ) , P o N ( x o + t { wo + w,[wol})).

Indeed, on the sphere aD, of radius R (chosen sufficiently large), the uniform boundedness of N and Hypothesis (A) imply once again that if h ( x , t ) = 0, both 11u11 and ((xollmust be small, so that h ( x , t ) # 0 on do,. Finally, we observe that the essentialness of homotopy class of go(w) is unaffected by assuming that L is the identity mapping since L is here regarded as a linear homeomorphism of V onto L ( V ) . Consequently the go(w) can be written go(w) = ( u , P,N(Ra)) where a E { a I a E Ker L , llall = l}. Thus the homotopy class of the associated normalized map 8, = go/]go!,[go] coincides with the mth iterated Freudenthal suspension of i;, E m[ 61. Consequently the desired result is established. Remarks: (a) The result (5.3.22) does not require the compactness of N for its validity. This is another significant improvement over (5.3.21). (b) The result (5.3.22) is sharp in the sense that it is not difficult to construct two examples, (i) a nonsolvable equation Lu + Nu = 0, where L and N satisfy the hypothesis of Theorem (5.3.21) and the homotopy class of [ ii]is nontrivial, and as well another (ii) a solvable equation with the stable homotopy class of [ ji] trivial, but the homotopy class [ ,ii] itself nontrivial. Here we sketch the fundamental ideas involved in (i) choosing the Hilbert space H to be the sequence space 1, with the standard inner product and L N E M ( H , H ) . Actually we seek the desired example in the simplest possible case index L = I , and attempt to exploit the interesting homotopy facts that r,(S2) = Z, while r4(S3) = Z , where the Freudenthal suspension of a generator a of r3(S2),E [ a ] # 0 while E [ 2 a ] = 0. Thus if the mapping N is so chosen that the homotopy class of the associated mapping 6, [ j i ] = [ 2 a ] E n,(S2), then [ p ] # 0 while E K [b] = 0 for K > 0. To be more precise, we let x = (x,, x2, . . . ) denote a typical element of 1, and set Lx = (0, 0,0,x,, x6, x,, . . . ) so that dim Ker L = 4, dim coker L = 3 and index L = 1. To define N we note that in terms of the complex numbers Z , , Z , a representative h ( Z , , Z,) of [ 2 a ] is given by h ( Z , , 2,) = ( 2 Z ~ ( Z , / - ' Z 2I ,- 21Z,I2) and a representative + = (+,, +,, +,) of E [ 2 a ] is given for real X, by + € ( Z , , Z,, X,)= ( h ( Z , , Z,), 2X,/Z,I). Moreover, since-+€ is_inesse_ntia_l,it is has +3, +,). Morenonzero extensio? to the interior of S 5 which we denote by + = (+,, over, we extend + to R by setting + ( R a ) = &(a) for R > l. Maintaining the notation of (5.3.21) for ge_neric u E I,, we set u = r a + x, r a E Ker L, (la(l= 1, x l Ker L, and define N,(Ra x) =+,(a) ( i = I, 2, 3). N,(Ra + x), = +,(u) - a, are N,(u)= 0, i > 5. One easily verifies that N = (N,, N,, N,, N,, . . . ) satisfies Hypothesis (A), and as mentioned above one easily shows that the mapping associated with N is such that [ ji] # 0, while the stable homotopy class of E [ ji] = 0. Finally, one easily shows that the equation Lu Nu = 0 is not solvable in 1,. Indeed Lu + Nu # 0 by the construction of +. On the other hand, if R > I , th_e fourth coordinate of Lu + Nu is a,(2/Z2/ + R - 1) > 0 for a, # 0; whereas if a, = 0, + = (+,, q2,+3, 0) # 0 for (al, a,, a,, 17,) E S3. (c) Operators L + N as discussed in (5.3.21) arise by expanding a nonlinear Fredholm operator about a singular point.

+

+,,

+,,

+

+

5.3 HOMOTOPY. THE DEGREE OF MAPPINGS. AND ITS GENERALIZATIONS

263

5.3E Generalized degree for C2 proper Fredholm operators of index zero

If the mapping f ( x ) - p belongs to ?:(do, X ) and is smooth (C2 say) on the bounded domain D of X, then its Leray-Schauder degree can be defined by differential techniques. More precisely, assuming f ( x ) # p for x E aD, we shall show that the Leray-Schauder degree d( f,p,D ) can be computed as follows: Step 1: Suppose that on the set up of all solutions of f ( x ) = p in D, Y(x) is invertible. Then by the inverse function theorem and the fact that f is proper on D, up is finite; and we set

(5.3-24)

d ( f , p , D ) =.

x

4 f ’ ( x ) ,0, D).

x E up

Step 2: If f ’ ( x ) is not invertible on up, by (3.1.45), we may find a sequence pn+ p in X so that f ( x ) # p , on dD, and on the set up” = { x I x E D ,f ( x ) = p , , } , f ’ ( x ) is invertible. We then set d ( f , p , D ) = limn+a, d(f>p,,, D).

(5.3.25) The definition of d ( f , p , D ) just given coincides with the definition given in Section 5.3C. Proof: If f ’ ( x ) is invertible on up, the two definitions certainly coincide by virtue of the properties of Leray-Schauder degree. Otherwise, we observe that since the Leray-Schauder degree d(f,p , D ) is continuous in p and f ( x ) # p on aD, limn-tmd(f,p,, D ) certainly exists as pn + p and equals d(f, p , D). The method of defining the degree of smooth mappings described above clearly applies to a very broad class of (nonlinear) Fredholm operators. However, in any such extension of the Leray-Schauder degree, if homotopies through proper Fredholm operators of fixed index are allowed, the crucial property of homotopy invariance cannot be preserved without additional restriction. For example, Kuiper proved that the group of linear invertible operators defined on a separable infinite-dimensional Hilbert space H is contractible. Thus any two invertible operators L , and L, defined on H are homotopic on the unit sphere {IlxllH = l } through invertible linear Fredholm operators (of index zero), even though with any definition of the type (5.3.24), L , and L, may have different signs. Hence, for proper smooth Fredholm operators of index zero defined on D,we shall define a homotopy invariant (mod 2) degree in the following manner: Let f be a proper Fredholm operator of index zero and class C 2

264


on a bounded domain D with values in a Banach space Y , and suppose f ( x ) # p for x E a D .

Step 1’: Suppose that at each point of the set up s { x 1 x E D , f ( x ) = p } the mapping! is regular (i.e., f ’ ( x ) is a surjective linear mapping of X + Y ) , then the properness off ensures that the set up is compact, while the fact that f ’ ( x ) is of index xero and surjective implies that f ’ ( x ) is invertible and hence that f ( x ) is a local homeomorphism on up, by the inverse function theorem. Consequently, up is finite, and we set the generalized degree d,(f, p , D ) equal to the parity of the set up. Step 2’: If at some point of up the mapping f is not regular, by (3. I .45), we may find a sequencep, + p in Y such thatf(x) # p , for x E a l l and on up” = { x 1 x E D , f ( x ) = p , } f is regular. Then we set d,(f, p , D ) = limn+md,(f,p,, D). Of course, the definition just given is sensible only if d , ( f , p , D ) is independent of the sequence p , and if the relevant limit exists. In fact, we prove (5.3.26) The function d,(f,p, D ) discussed in steps l’, 2‘ is well defined. Proof: It suffices to show that if p is a regular value for f and f ( x ) # p for x E aD, then the number of points in f - ’ ( p ) n D is a locally constant function in C 2 ( D )n C ( B ) .For then, if there are two sequences of regular values fory, { p , , } and {q,,}, both tending t o p in Y , for n sufficiently large, d,(f, p , , D ) = d , ( f , q,, D ) . In addition, the sequence of integers d,(f, p,, D )stabilizes. Consequently, the definition of d,(f, p , D )in step 2’ will be justified.

We prove a slightly more general result that if f ( x ) # p for x E aD, p is a regular value for f, and g is sufficiently close to f in C ’ ( D )n C ( a D ) , then the number of points in f - ’ ( p ) n D is equal to the number of points in g - ’ ( p ) n D. As discussed in step 1’ abovef-’(p) n D contains a finite number of points x l , . . . , xk (say). Let 0; ( i = 1 , . . . , k ) be a family of small pairwise disjoint open neighborhoods with x, E O i . Then f(0- Uf=,Oj) does not contain p ; and for g sufficiently close to f, the also does not contain p . Since properness off implies that g ( D - U ,Oi) f ’ ( x j )is a surjective linear Fredholm operator of index zero ( i = 1, . . . , k ) , f ’ ( x j ) is a linear homeomorphism for each i = 1, . . . , k . The inverse function theorem then implies that g is a diffeomorphism of 0; onto a neighborhood of p . Thus there is exactly one point zi E 0;with g ( z J = p . This means that the number of points in f - ’ ( p ) n D is the same as the number of points in g - ’ ( p ) n D.

:I

Now we show that the function d ( f , p , D ) has the crucial properties of degree.

5.3

HOMOTOPY, T H E DEGREE OF MAPPINGS, A N D ITS GENERALIZATIONS

265

(5.3.27) Suppose that f is a C' proper Fredholm mapping of index zero defined on D (a convex open subset of X ) withf(x) # p on aD. Then: (i) d,(f, p , D ) # 0 implies that the equation f ( x ) = p has a solution in D (so if f ( x ) # p in D,d g ( f ,p . D )= 0); (ii) d,(f,p, D ) is invariant under proper C z homotopies h ( x , r ) that are Fredholm operators of index zero, with h ( x , t ) # p for x E aD, t E [O, I]; (iii) dg(f,p , D ) is continuous in p , and f E C 2 and depends only on the component in Y - f ( a D ) that containsp. (iv) If D is a ball with center at the origin, and f is odd, d(f, 0, 0 ) # 0. If d(f,p . D ) # 0, by definition, there is a sequence of points D such that f(x,) = p,,. Sincef is proper on 5, { x,,} has a convergent subsequence with limit 3 . Clearly the continuity of f implies that f(Z) = p and X E D since f ( x ) # p for x E dD. If f ( x ) # p in 0,then d g ( f ,p , D )= 0, by definition. (ii): Suppose h ( x , t ) is a C z proper Fredholm operator of index zero defined on 5 X [0, 11joiningfand g, such that h ( x , 1 ) # p for x E i3D and t E [0, 11, and p is a regular value for h ( x , t ) . Then h - ' ( p ) is a compact one-dimensional manifold' with boundary equal to ( f - ' ( p ) , 0) u ( g - ' ( p ) , 1); i.e., the number of points in f - ' ( p ) (denoted # ( f - ' ( p ) ) ) and in g - ' ( p ) (denoted #( g - ' ( p ) ) > . Since the boundary of a compact onedimensional manifold has an even number of points. Proof: (i):

p,, + p and

x,, E

(5.3.28)

#(f-'(p))

=

#( g - ' ( p ) ) (mod 2).

Suppose now that p is regular for f and g but not for h , then by the argument in the proof of (5.3.26), there is a neighborhood V of p such that for all p' E V . #(g-'(p')) =#(g-'(p)) By (3.1.45), there is a regular value j for h ( x , t ) in V , and since (5.3.28) holds for p" by our first argument, it holds also for p . Finally, if p is not regular for either f or g, by (3.1.45), there is a sequencep, + p that will be regular for both and such that h ( x , t ) # p,, on

#(f-'(p'))

=#(f-'(p));

aD x [0, I]. Consequently, by the definition of d ( f , p , 0 ) and the above paragraph,

d , ( f , p , 0) = d,(f,p,. D )= q g , p , , , D )= d,(g.p, 0). (iii)-(iv):

These are immediate consequences of (ii), as in Section 5.3C.

For fixed t, h ( x , 1 ) is a Fredholm mapping of index 0. Since h ( x , f)is C 2 and proper on D x [0, I], h ( x , f)is Fredholm of index 1, and so at a regular valuep of h ( x , 1). dim h - ' ( p ) = 1.

266


5.4 Homotopy and Mapping Properties of Nonlinear Operators

In this section, we shall derive general mapping properties of a nonlinear operator f E C ( X , Y ) from facts about (a) (generalized) degree functions relative to various bounded domains D c X , and more generally (b) the "essentialness" off relative to D . Unless otherwise stated, we shall restrict the class of admissible mappings A discussed here by assuming that if D is a convex bounded domain of X with f ( x ) # p on aD, then there is a degree function d ( f , p , D )(with values in Z or Z,) such that:

+

(i) a(j,p , D ) o implies p E ~ ( D I ; (ii) d(f,p , D ) is invariant under admissable compact homotopies; and (iii) if D is a ball with_ center at the origin and f is odd, then f is essential relative to D and d(f,0, D ) # 0. By our discussion of the preceding section, A includes compact perturbations of the identity and linear Fredholm operators of nonnegative index, as well as, by Section 5.3E, compact perturbations of proper C2 Fredholm operators of index zero. 5.4A Surjectivity properties

We first prove (5.4.1) Theorem Suppose f E C ( X , Y ) n A is proper and there is some point p* E Y such that d ( f , p * , Z) # 0 whenever Z is an open ball centered at the origin containing f-l(p*). Thenf is surjective. Proof: Let p E Y and let L be the straight line segment in Y joining p and p * . Then since f is proper and L is compact, f - ' ( L ) is compact and consequently bounded. Hence we can find a ball 2 , = { x 1 ( ( ~ ( 1< R ] with sufficiently large radius R so that f-'(L) c Z,. Then if we denote the points of L byp(t) = zp (1 - t)p* f o r t E [0, I], the homotopy invariance of degree implies

+

(5.4.2)

2(f7p,

ZR>

= d"(f,p(l),

c,)

=

Z),

d"(f,p*5

Thus the equation f ( x ) = p has a solution in Z., subjective.

# O.

Consequently f is

(5.4.3) Corollary Let f E C ( X , Y ) n A be an odd proper mapping, then f is surjective.

Proof: Sincef E

X and is proper,f-'(O) is bounded, so that d(f,0, Z),

5.4 HOMOTOPY A N D MAPPING PROPERTIES OF NONLINEAR OPERATORS

267

# 0 for the ball 2, = { x 1 IIx(J< R } with R so large that f -'(O) c 2,. Thus f is surjective by the above Theorem (5.4.2). (5.4.4) Corollary Let f E C ( X , Y ) n A be a proper complex analytic mapping between complex Banach spaces. Suppose (as is the case for the Leray-Schauder degree) that for such mappings, d ( f , p , D) # 0 whenever D is a ball and p E f(0)- f ( d 0 ) . Then f is surjective. Proof: By Theorem (5.4.2) it suffices to find a point p * E Y such that d ( f , p * , Z) # 0 whenever the ball C is so large as to containf-'(p,). Now ) let p * _be any point in f ( X ) . Then, by hypothesis, since p* E ~ ( Z f ( d X), d(f,p , , C) # 0. Consequently, f is surjective.

(5.4.5) Corollary Suppose C is a compact, asymptotically linear operator defined on a Banach space X with asymptotic derivative C,. Furthermore, suppose L E L ( X , Y ) is a linear Fredholm operator of index zero such that L C , is invertible. Then f = L + C is surjective.

+

Proof: Under the given hypotheses, we first prove that i f f = L + C, then the inverse image of a bounded set B of Y is bounded in X , and that f is a closed mapping. Then, by repeating exactly the argument given in Theorem (5.3,16(ii)) with d(f - p , 0, D ) the Leray-Schauder degree, we prove that f is surjective. First, we prove that i f B is a bounded set in Y , thenf-'(B) is bounded in X . Otherwise there would be a sequence x,, E X with (Ix,(I 400 and a number M independent of n such that (5.4.6)

ll(L + C,)X, + ( C - C,)X,II

< M.

On the other hand, since L + C , is invertible, there is a constant k > 0 (independent of n) such that ll(L + C,)x,II > kllx,ll. Therefore, (5.4.6) implies

Letting n + 00, we have the desired contradiction, since C , is the asymptotic derivative of C and consequently Il(C - C ' , ) x , ~ / / ~ ~<xi, k~ ~for n sufficiently large. Next we prove that f is a closed mapping. Indeed, let C be a closed set in X , yn E f (C) with yn in Y . By the above argument there is a bounded set { x,} such that f ( x , ) = y,. Thus after possibly passing to a subsequence, we may suppose that Cx, .+ z (say), so that { Lx,} converges strongly in Y . But, by (1.3.37), some subsequence {x,,) converges to 2 (say). By continuity, f(2)= y . Thus f has closed range. ?!+

268

5


5.4B Univalence and homeomorphism properties

The degree can be used to prove global univalence results from local data, as in Section 5.1. As an example, we prove the following uniqueness result .

(5.4.7) Theorem Let D be a bounded domain of a Banach space X and

f a local homeomorphism of D+ X . If f E (?, ( a D , X ) is a compact perturbation of the identity with d(f,p , D )= & I , then the equation f ( x ) = p has precisely one solution in D . Proof: First we observe that since f is a local homeomorphism and f is proper on 5, the set up = { x 1 f ( x ) = p , x E D } is discrete and hence finite. By (5.3.24),

(5.4.8)

* 1 = d(f,p,D ) =

2

d ( f , p , 0,)

xEf-'(P)

where the 0, are small disjoint open sets containing x. Thus it suffices to show that (*) d ( f , f ( p ( t ) ) ,Opct))= constant whenever p ( t ) is a path in D. For then, d ( f , p , 0,) = d ( f , p , 0,) if x , y E up; so that by (5.4.Q the number of points in up is one. To prove (*), let p ( t ) be any path in D. Then for fixed T , let Opc0,be chosen as above and let p ( t J E Op(T).Then d ( t ) = d ( f , f ( p ( t ) ) , OPcT,)is a constant function of t E [ t , , TI, by the homotopy invariance of degree. We shall prove that d(t) = d ( f , f ( p ( t ) ) , Opt,)) is a constant function of t E [ t , , 7'1. To this end, let Spc,,)be an open ball about p ( t J contained in Op(t,)n O p ( T ) . Then

Thus provided It, - TI is sufficiently small to ensure that p ( t J c Op(T), (5.4.9)implies that d ( t ) is constant in [ t , , TI. Hence the set { t 1 d ( t ) = d(1)) is open and closed in [0, 11, and therefore d ( t ) is a constant function of t E [0, I], as required.

(5.4.10) Corollary Suppose the hypotheses of (5.4.7)are satisfied and that f(0)n f ( a D ) = 0. Thenf is a univalent mapping of D onto f ( D ) . Proof: To prove the result, we note that f(0)c ( X - f ( a D ) ) , so that d(f,p', D ) is defined for p' E f ( D ) and is constant since f ( D ) is arcwise connected. Thus by hypothesis d(f,p', D ) = 1 ; and by Theorem (5.4.7) the equation f ( x ) = p' has exactly one solution. Therefore f is univalent on

*

f(D).

In view of the foregoing, it is natural to ask whether a one-to-one mapping f of a bounded open set D onto f ( D ) is indeed a homeo-

5.4 HOMOTOPY A N D MAPPING PROPERTIESOF NONLINEAR OPERATORS

269

morphism. As in the finite-dimensional case, any such result requires a considerable proof. Actually we prove (5.4.1 I ) Invariance of Domain Theorem Let f E C ( D , Y ) n A be a compact perturbation of a linear mapping L that is a Fredholm operator of zero index, where D is an open subset of the Banach space X . I f f is a one-to-one mapping of D ontof(D), then f is an open mapping and hence a homeomorphism of D ontof(D).

Proof: We prove that an interior point xo of D is mapped by f into an interior point of f ( D ) in two steps. First, we show that under the given hypotheses, if the mappingf(x) is essential in a small ball C about xo, then for some z > Of(Z) contains a sphere of radius z aboutf(x,). Secondly, we show that under the hypotheses of the theorem, f ( x ) - f ( x o )is essential in Z by proving that f is homotopic to an odd mapping S E A and by using property (iii) defining A. Without loss of generality, suppose xo is the origin and the radius of Z is one. Then to complete the first step of the proof just outlined, we observe thatf(i3C) is a close$ set sincefis proper on bounded subsets of D, so that the distance c = d(f(a Z), 0) > 0 since f ( x ) = f ( x ) - f(0) # 0-on 3 2 . Now let IIy -f(O)ll < c, then we show that g ( x ) = f ( x ) - y and f ( x ) are compactly homotopic on ax, by setting h ( x , t ) = f ( x ) - 9 - (1 - t)f(O) for t E [0, 11. Then on aZ, Ilh(X3

r)ll

=

Ilf(.> -fPll

+ tllf(0) -

tvll > c -

0.

By (5.3.3). sincef(x) is essential on aZ by hypothesis, g is also essential on and so f ( x ) = y has a solution in C for IIy - f(0)Il < c. Consequently f(Z) covers an open c-sphere about f(O), and the first step is completed. Next we show that under the hypotheses, the mappingf(x) = f ( x ) - f(0) is essential relative to 2. We do this by showing that f is compactly homotopic to-an odd mapping j E t7c. Indeed, i f f = L C with C compact, let h ( x , t ) = Lx + { C(x/(l + t ) ) - C ( - t x / ( l + t ) ) } . Clearly h ( x , 1 ) is the desired compact homotopy since on aZ, if h(xo, to) = 0 for llxoll = 1 and some to E [0, 11, thenf(x/(l + t ) ) = f ( - t x / ( l +,f) which is not possible since f is one-to-one on Z. Now h ( x , t ) joins f and the odd mapping h ( x , 1) = Lx - { C(x/2) - C( x/2)}, which is essential on Z since h ( x , 1) E %. Thusf is essential on Z, as required.

a Z,

+

5.4C Fixed point theorems As was mentioned in Chapter 3, it is often important to give precise conditions under which a mapping f of a Banach space X into itself possesses a fixed point. By virtue of example (5.3.1), the direct extension of

2 70


the Brouwer fixed point theorem (1.6.4), based only on the continuity off, is false. Consequently, it is natural to attempt to solve the equation x = f ( x ) by supposing that the mapping I - f E A. As a first result we prove the following version of Schauder's fixed point theorem (2.4.3). (Rothe) Letf be a compact mapping defined on the closed unit ball % = { x I /\xll < I } of a Banach space X . Suppose f maps 32 = { x I I ( x (= / 1 ) into %. Then f has a fixed point in

(5.4.12)

c.

Proof: Suppose not. Then, by virtue of the homotopy invariance of the Leray-Schauder degree, since the compact homotopy h ( x , t ) = I - tf # 0 fortE[O, I l a n d x E 3 2 , d ( I - f , 0, Z) = d ( h ( x , f ) , 0, X) = d ( I , 0, 2 ) = 1.

Consequently, the equation x = f-( x ) has a solution in Z, contradicting the fact that f has no fixed point in Z. For complex analytic mappings f, (5.4.12) can be considerably sharpened, as follows. Corollary' Suppose that f is a compact complex analytic mapping defined on the closed unit ball %, of a complex Banach space X . In addition, suppose thatf maps a Z , = { x 1 /Ix(/= 1 ) into the interior of 2,. Then f has one and only one fixed point in 2,. (5.4.13)

Proof: The proof of (5.4.12) above shows that d ( 1 - f, 0, 2,) = 1. Thus the result (5.4.7) implies that the fixed point xo off must be unique.

The argument just given can be easily extended to prove the following a priori bound principle.

Theorem Let f(x, t ) be a one-parameter family of compact operators defined on a Banach space X for t E [0, I], withf(x, t ) uniformly continuous in t for fixed x E X . Furthermore, suppose that every solution of x = f ( x , t ) for some t E [0, 11, is contained in the fixed open ball Z = ( x 1 / / x i /< M ). Then, assuming f ( x , 0) G 0, the compact operator f ( x , 1) has a fixed point x E Z. (5.4.14)

Proof: Sincef(x, t ) is compact for fixed t on and uniformly continuous in t for fixed x , f ( x , t ) is compact on X x [0, 11. Furthermore, h ( x , t) = x - f ( x , t ) is a compact homotopy on = { x 1 IJxIJ = M ) , since by assumption, x # f ( x , t ) ror x E a2 and t E [0, I]. Thus by the homotopy invariance of degree, A c e by hypothesis the equation x = . f ( t , x) has no solutions on a2 for any t E [0, 11, d ( x - f ( x , I ) , 0, Z) = d ( h ( x , t ) , 0, 2 ) = d ( ~0,, 2 ) = 1.

'

Earle and Hamilton have shown that the compactness hypothesis in this result may be removed.

5.4 HOMOTOPY A N D MAPPING PROPERTIES OF N O N L I N E A R OPERATORS

271

Consequentlyf(x, 1) has a fixed point in Z. In the same circle of ideas, we prove

(5.4.15) Let f be a compact mapping of the closed unit sphere = { x 1 llxll < l } into the Banach space X such that the mapping g = I satisfies (5.4.16)

g(x) # ,8g( - x)

Then f has a fixed point in

for any ,8

>0

and

x E

E, -

f

ax,.

El.

Proof: Again, suppose f has no fixed point in

El. Then setting

we note that the hypothesis (5.4.16) above implies that g,(x) # 0 for = g. Thus d(g,, 0, 2 , ) for t E [0, I ] is defined, and by virtue of the odd parity property of g, of the Leray-Schauder degree, d( g, 0, Z,) = odd integer. Consequently, by the homotopy invariance of degree d ( g , 0, Z,) = d(gI,0, C,) # 0, g(x) = 0 is solvable in Z,, so that f has a fixed point in Z,. This - fact, however, contradicts the assumption that f has no fixed points in S,, and so the proof is completed. x E Z,, and that g,(x)

Another interesting result in this connection is the following analogue of (5.4.12).

(5.4.17) Let f be a compact mapping defined on a bounded domain D, such that aD does not contain the origin of a Hilbert space X . Moreover, suppose that (5.4.18)

(f(.x).

Then f has a fixed

< JJxJI’ point in 0.

x)

for each x E a D .

Proof: Suppose f has no fixed point in 5, then the mapping g = l - f E 2,(do,X ) and the Leray-Schauder degree d ( l - f, 0, D )= 0. Consequently g cannot be compactly homotopic to 1 on aD. Thus for some A, E (0, I ] and x, E aD. x, = X,f(.x,). But (5.4.18) implies that A, > I . Hence A,, = I , and so f has a fixed point on aD. This is the desired contradiction, and the theorem is established.

An interesting Banach space analogue of (5.4.17) is

(5.4.19) Let T be a compact mapping in C ( D , X ) , where D = { x I / J x J I < I}. Furthermore, for each x E aD, /Ix - TxJI’ 2 )I Tx1I2- IlxII’. Then T has a fixed point in 0. Proof: We consider the compact homotopy h ( x , t ) = x - tTx defined on

212


aD X [0, 11, and repeat the argument of (5.4.14), assuming T has no fixed point in 5 and obtaining a contradiction by showing that h ( x , t ) # 0 on aD X [0, 11. Indeed, if x, E do, to E (0, I), and h(x,, to) = 0, then

implying that IITx,ll2 -

(JxoI12 =

l / t i - 1 = (1

-

ti)/r,”.

Thus the hypothesis of (5.4.19) implies that (1 -

t,fII

Tx,112 2 ( I -

(1 - to)Il T X , ( ~ ~ ( I (1

-

to)[t i ( /Tx,l12] > 1

t,’)t;2

+ to)tg2 + to

(since to # 1)

(an impossibility).

Thus we have obtained the desired contradiction and the result is established. 5.4D Spectral properties and nonlinear eigenvalue problems

Suppose f ( x , A) is a one-parameter family of compact perturbations of the identity defined on 5 x R’and depending (uniform) continuously on the real parameter A, where D is a domain of a Banach space X . Furthermore, suppose that f(0, A) = 0. Then fhe Leray-Schauder degree can be used to great advantage in the study of solutions (x, A) of the equation f ( x , A) = 0 other than the obvious “trivial” solutions (0, A). As a simple example, we prove (5.4.20) Theorem Suppose that the one-parameter family f ( x , A) satisfies the above restrictions, and that for two distinct values of A, A,, and A,, the Leray-Schauder degree is defined and d ( f ( x , A,), 0, D ) # d ( f ( x , A,), 0, D).Then the equation f ( x , A) = 0 has a solution (F, with X E dD and E [A,, A,].

x)

Proof: Assume that the equation f ( x , A) = 0 has no solution (F,h) with E [A,, A,]. Then h ( x , t ) = f ( x , t A , + ( 1 - t)A,) for t E [0, 11 defines a compact homotopy joining f ( x , A,) and f ( x , A,). By the homotopy invariance of degree, d ( f ( x , A,), 0, 0 ) = d ( f ( x , A,), 0, D), which contradicts the hypotheses of the theorem. Consequently, h(x,, to) = 0 for some x, E all and to E [0, I].

X E aD and

x

5.4 HOMOTOPY A N D M A P P I N G PROPERTIES OF NONLINEAR OPERATORS

273

As a simple but interesting consequence of (5.4.20) we mention (5.4.21) Corollary Suppose N is a compact, asymptotically linear operator with asymptotic derivative C, defined on a Banach space X . If A;' is an eigenvalue of odd multiplicity of C, then for any c > 0 there is a ball Z of X such that for every open set D containing Z, the equation x = ANx has a solution (F, A) with X E aD and A E [A, - e , A, €1.

+

Proof: Let E > 0 be given. Then we shall calculate the Leray-Schauder degree of the operators I - (A, + E ) N and Z - (A, - E ) N ,at zero, relative to any bounded set D of X that contains a ball Z, = { x I llxll < R,) with R, sufficiently large. We shall show that these degrees are different; so that by Theorem (5.4.20), the equation x = ANx has the type of solution (X, A) described above. To calculate the Leray-Schauder degree d(Z - (A, c ) N , 0, D ) , we shall r ) N is compactly homotopic on aD to the linear show that Z - (A, operator L, = Z - (A, + E)C, provided d ( a D , 0) is sufficiently large. Indeed, suppose € > 0 is sufficiently small, then since C is compact and L, is invertible, there is a constant p (independent of x ) such that IIL,xll 2 pllxll. Thus, since C is the asymptotic derivative of N , for t E [0, I ] and llxll 2 R, so large that IINx - Cxll < Pilxll/2(lAol + I),

+

+

IIL,x - (A, + E ) f ( N X

-

CX)ll

> IIL,xll

-

t(Ao + E)IINX - Cxll

{ P - tP)llxll= tPllxll

>o*

Hence, by the homotopy invariance of degree and (5.3.16), if D contains

=w (5.4.22)

d( Z - (A,

+ E ) N ,0, D ) = d( L,, 0, D ) = (

-

1)'

where p is the number of eigenvalues of C greater than (A,+E)-'. larly, if L - ,= Z - (A, - 6 ) C, then (5.4.23)

d(Z

-

(A,

- E ) N ,0, D )=

Simi-

d ( L - , , 0, D ) = ( - I)",

where p , is the number of eigenvalues of C greater than (A, - € ) - I . Since the multiplicity of A,' is an odd number, p # pl (mod 2), and so the Leray-Schauder degrees of I - (A,? € ) N on D relative to zero are different, as desired. Thus the corollary is established. In the same way, we prove (5.4.24) Corollary Suppose D is a bounded open set containing the origin in the Banach space X . Suppose N is a compact operator mapping aD into X with (5.4.25)

IINxII

>0

for each

x EaD.

274


Then the equation x

= ANx

has a solution (2,A) with X E aD.

Proof: Suppose the corollary is false. Then by (5.4.20), the function d(A) = d(Z - AN, 0, D ) is defined for all A E R', and is, in fact, a constant function. Clearly, for A = 0, d(A) = 1 since d(A) = d ( l , 0, D ) in that case. We shall obtain a contradiction, by using (5.4.25) to show that d(A) # 1 for some A of sufficiently large absolute value. (By virtue of (5.3.14), this fact is independent of any compact extension of N into D.) To this end, observe that the compactness of N implies the existence of a number a > 0 such that 11 Nxl( > a > 0 for x E aD;while the fact that D is bounded implies that (Ix((< R (say). Then for A sufficiently large, A > 2 R / a (say), [IAN - 111 > Aa - llxll > R. Thus by the definition of the Leray-Schauder degree, there is a finite-dimensional subspace X,, of X and a compact mapping N,,: D - X , , approximating N such that d(Z - AN, 0, D )= d(Z - AN,,, 0, D). Moreover, if the restriction of N,, to D n X , is denoted N,,, then d(Z - AN,,, 0, D )= d,(l - A i , , , 0, D n X,,). Without loss of generality we may suppose that dim X,, is odd, and that on aD n X,,, IIN,,xll > .4 Next, we demonstrate (*) Z ? AN, is homotopic to +AN, on aD n X,, (avoiding zero), for (A( sufficiently large. Once (*) is established, the theorem will be proved. The fact that X , i_s odd dimensional implies that (for A # 0) the Brouwer degrees dB(AN,, 0, D n X,,) and dB(-AN,,, 0, D n X,,) are either both zero or of opposite sign, so that one of them must be different from unity. Then by the conclusion of the last paragraph, for some A E (- 0 0 , co), d(Z - AN, 0, D ) # 1, as required. To prove (*), let 1 PI be sufficiently large and x E aD n X,,. Then for t E [O, 11, Ilt(x + PN,,x) + (1 - f ) P N , , X I I = 11 PN,X + txll

t

2 I PI IINnxII - IIxII > I PI. - R Thus (*) is established and the theorem is proven.

> 0.

For complex analytic mappings, we have the following important consequence of (5.3.16(v)). (5.4.25') Corollary Let D be a bounded domain of a complex Banach space X (with 0 9 aD), and let f ( x , A) be a one-parameter family of complex analytic mappings defined on D X R', which are compact on the product of 5 with any bounded interval of R',and such that f ( x , 0) = 0. Suppose, in addition, that (x,, A,) E D x Iw' is a point of bifurcation for the equation&x, A) = x - f ( x , A) = 0. Then the equation g(x, A) = 0 has a solution (X, A) with X on aD and E (0, A,]. Proof: Since (x,, A,) is a point of bifurcation of g(x, A) = 0, g,(x,

A,) is

5.4 HOMOTOPY A N D M A P P I N G PROPERTIES OF N O N L I N E A R OPERATORS

275

not invertible; and so by virtue of (5.3.16(v)), if g(x. A,) = 0 has no solutions on aD, then d( g(x, A,), 0. D ) > 2. However, d( g(x, 0). 0, D ) = 0 or 1 depending on whether or not 0 E D. In either case, the conclusion of the corollary follows from Theorem (5.4.20). We now use the results just established to study two related questions: (i) spectral problems for the equation x = f ( x . A), wheref(0, A) = 0, in which one studies the “spectrum” up of the solutions S of x = f ( x , A) as A varies over the real numbers, where up = { p 1 p E R1,(x, p ) E S x # 0); (ii) continuation problems for .Y = f ( x , A); in which one supposes (x,, A,) is a point of bifurcation (in the sense of Chapter 4) for the equation x = f ( x , A), and one studies the component of the closure of the nontrivial solutions (.?, h) E 5 that contains (x,, A,).

.

As a first result concerning the set up we let d ( Z , Y ) denote the distance between the sets Z , Y and prove

(5.4.26) Theorem Let f ( x , A) be a compact operator defined on X x ( - co, co) with f(0, A) = 0. and such that Ilf(x, A)ll+ co as A + 00 uniformly on every bounded set Z of X with d(Z, 0) > 0. Suppose that for every open set U containing the origin, the equation x = f ( x . A) has a solution ( x ( u ) , A,) with x ( u ) E aU and A,, E R l such that as Ilx(u)ii + co, A,-*A,, and as Ilx(u)li +O, A,+Ao. Then the equation x-= f ( x , A) has solutions (2,h) with x # 0 for any h E (A,, )A, - {0}, i.e., A E up. Proof: Suppose p E (A,, A,) - ( 0 ) is not in the set up. Then we shall obtain a contradiction by constructing a bounded open set V containing the origin but such that g(x, A) = x - f(x, A) has no nontrivial solution on aV. To this end, let the two components of R ’ - { p ) be E , and E,, where A, E E , and A, E E,. Moreover, suppose F , = ( x u 1 x,, = f ( x , , A,), A, E E m } and F, = { x u I x u = f ( x , , A,), A, E E,} u ( 0 ) . Clearly the compactness off(x, A) implies that the disjoint sets F , and F, are closed, while F, u F , contains all the nontrivial solutions of x = f ( x , A) mentioned in the theorem. Thus d(F,, 0) > 0, while the elements in F, are uniformly bounded. Next we can show that d(F,, F,) > 0. Indeed, otherwise there would be sequences (x,,) E F , and { y n } E F, that are uniformly bounded away from zero and infinity, but such that IJx, -ynll+O, while x, = f ( x , , A,) and y , = f ( y n . A;) for A, E E,, A; E E,. By hypothesis, we may then suppose that JA,] and IAAl are uniformly bounded, and after possibly passing to subsequences, we may suppose A, -+i and A; Hence, again possibly after passing to subsequences, we may assume that {x,}, and consequently {y,,}, converges strongly to z # 0, and that z E F, n F,. +XI.

276


This last fact is the desired contradiction, and consequently there is a number /3 > 0 with d(Fo, F,) = p. Now let O(Fo) be the bounded open set obtained as the union of open balls of radius 4 p centered at F,. Then dO(Fo) is disjoint from Fo and F,, contradicting the hypothesis of the theorem, so that p E (Ao, A,) - ( 0 ) . Next we turn to the continuation problem mentioned in Section 4.1 and prove the following global analogue of (4.2.3) for the equation ( I - AL)x + g(X, A) = 0. (5.4.27) (5.4.28) Theorem (Rabinowitz) Suppose L is a linear compact operator mapping of a Banach space X into itself with A,' an eigenvalue of odd multiplicity, while g(x, A) defined on a domain D X V of X X R ' is continuous and compact in x, continuous in A and of higher order in x at the origin, i.e.,

A)ll = o(llxll> as llxll +o uniformly for bounded A. Then, if denotes the closure of the component C of nontrivial solutions of (5.4.27) containing (0, Ao), one of the following alternatives hold: either (i) is noncompact in D x U (so that if D X R' coincides with X X R', is unbounded) or (ii) contains at least one but at most a finite number of points (0, A,) with A,-' an eigenvalue of L distinct from A, and the number of such Xi of odd multiplicity (excluding A,) must be odd.

I1

d . 7

c

c c

c

c

Proof: Suppose is compact in D X U . Then the number of distinct points of the form (0, A,) (as described in the theorem) must be finite since otherwise the compactness of L would imply the noncompactness of Thus to prove the theorem we need only prove the evenness of the number - points (0, A,) with A; an eigenvalue of odd multiplicity for L contained of C. To this end we choose a bounded open subset Q of D x U containing and such that there are no solutions (x, A) of (5.4.27) on a Q and so that Q contains no points of the form (0, A,) that differ from those of Then to measure the nontrivial solutions of f ( x , A) = ( I - AL)x g(x, A) on llxll = p, we consider the Leray-Schauder degree at (0, 0) of the mapping f p ( x , A) = ( f ( x , A), llxl12 - p 2 ) relative to Q. By our construction of Q this degree d, = d( f,, (0, 0), Q ) is defined. We shall obtain the parity result desired in three simple steps: (i) d, is

c.

'

c

+

c.

independent of p by the homotopy invariance of degree and is fact equal to 0

as one can choose p so large that f p ( x , A) = 0 has no solutions; (ii) then by choosing p small and showing the fact that the only contributions to d, for p small came from local contributions near points of the form (0, A,). This follows from (5.3.14); and finally (iii) computing d( f,,, (0, 0), Q ) for p smaN,


277

which we assert equals the right-hand side of (*) below, so together with (i) and (ii) we find

(*) 0 =

c {d(l 2

= Ak

- (A,

-

E)L, 0, llxll

< P ) - d ( l - (A,

E)L, 0, (Ix/l< P ) }

+2

(odd)

(where summation occurs only over A, of odd multiplicity). From which we conclude the parity of the points A, of odd multiplicity is even. To prove (iii) we compute the contribution to d, near each (0, Ak) d,(k), for small E # 0 and small p we consider the homotopy h ( x , t ) = t f , , ( x , A + € ) + (-l t ) { 1 - ( ( X , + E ) L , E i - E 2 }

+

relative to (0, 0) on the set S = ((x, E ) I llx112 c2 < p2 + €02). Clearly h ( x , t ) # 0 on ax, provided ( r , c) is chosen sufficiently small for in such a case E = 2 c0 and so x = 0. Thus by the homotopy invariance of degree d,(k) = d ( l - (A, + E ) L ,E; - c 2 ) . To compute this latter degree we use (5.3.16) since the only solutions of h ( x , 0) = 0 are p = 0, E = & c0, and the Frechet derivative of h ( x , 1) at (0, E ) is given by h’(0, E ) [ ; , Z ]

=

( ( I - AL)x,

-

2~:).

Thus at E = 2 c 0 the local index is - d ( l - (A, + E ) L ,0, llxll 0,

5.4E Necessary and sufficient conditions for solvability and its consequences

In the case of operator equations of the form Lu + Nu =f, where L E L ( X , Y ) is a linear Fredholm operator of nonnegative index p mapping a Banach space X into a Banach space Y , and N is a compact mapping that satisfies the hypotheses of (5.3.21). we can sharpen our previous results. lndeed we now prove (a) necessary and sufficient conditions for solvability and (b) the openness of the range of L + N . Indeed we begin with the index zero case with dim Ker L > 0 and demonstrate: (5.4.29) Theorem Suppose L is a linear Fredholm self-adjoint operator mapping a Hilbert space H into itself, and that Nu is a uniformly bounded compact continuous mapping of H into itself, such that the following limit exists uniformly for ( ( x / uniformly ( bounded, $(a) = limr-tm PoN(ra + x ) , a E Ker L n { Ilxll = 1). Moreover suppose for all positive r,

(5.4.30) ( N ( r u + x),

0)

< ( @ ( a ) ,u )

x l K e r L.

278


Then (i) a necessary and sufficient condition for the solvability of the equation Lu N u = f is that (f, a ) < (+(a),a). and (ii) the mapping L N has open range.

+

+

Proof: The necessity of the condition (f, a ) < ($(a), a ) follows immediately from (5.4.30) by taking the inner product of Lu N u = f with a E Ker L and using the self-adjointness of L. To derive the sufficiency of the condition we first observe if it is satisfied, Hypothesis (A) of (5.3.21) is satisfied for the operator L + N via (5.4.30), and moreover on a sufficiently large sphere ax, of radius r in Ker L , the Brouwer degree dB(+(u),f. Z,) = I . Consequently the criterion of (5.3.21) implies f E Range(L + N ) . To derive the openness of the range of L N we show that if fo E Range(L N ) , the range of L N also contains a ball of positive radius about f. This end we argue as follows. First if f - fo E (Ker L ) I the solvability of the equation Lu N u = f is an immediate consequence of part (i) just established. On the other hand, if the projection o f f - fo onto Ker L is sufficiently small in norm, f is also in the range of L + N , by virtue of the strict inequality of the necessary and sufficient condition (as stated) and the finiteness of dim Ker L. Thus (5.4.29) is established.

+

+

+

+ +

As an application of this result we give an alternative proof of the solvability of the partial differential equation already discussed in Theorem (5.1 A), concerning negative constant Gaussian curvature metrics on compact two-manifolds M . The equation involved can be written: (5.4.31)

Au - eZu= K ( x ) ,

with

vol(

a, g)= 1

where A denotes the Laplace-Beltrami operator defined on the manifold do not seem to be satisfied by (5.4.31) since the nonlinear term exp 2u is not uniformly bounded in any Banach space so far mentioned. T o overcome this difficulty we apply the maximum principle for A on ‘X, which states that if u ( x ) is a smooth solution of (5.4.31), then at a positive maximum xo of u ( x ) , Au(xo) = exp 2 u ( x 0 ) + K ( x o ) < 0 so that u ( x o ) < co (co an absolute constant). This justifies our replacing the equation (5.3.3 1) by

(a, g). Clearly the hypotheses of (5.4.29)

(5.4.31’)

Au - f0(u)

= K(x),

wherejo(u) = eZufor u < co andfo(u) strictly increasing with limitf(oo) for u 2 c,,. Now fo(u) is uniformly bounded and we can give an alternate proof for the result (5.1.8). A necessary and sufficient condition for the solvability of (5.4.31) is that < 0, i.e., the mean value of K ( x ) over ( M , g ) be negative. Proof: Clearly it suffices to consider only the truncated equation


279

(5.4.31'). This equation can clearly be written in the Sobolev space W,,2( g) in the form

a,

LU + N U = - g , where L is the operator naturally associated with A, N u with f $ u ) and g with K ( x ) . Here L is self-adjoint and (5.4.30) is satisfied with ( L k u ) = JqKl

v u12.

( N u , G ) = JJJu)t'.

K ( x ) u = (8,G),

L l

Ker L consists of the constants. Thus (5.4.31') is transformed into operator form and the result (5.4.29) is applicable. We observe that the solvability criteria of (5.4.29) becomes for a = - 1 1) = 0.

Thus the necessary and sufficient conditions for solvability are satisfied provided the constant co in the definition of f o ( u ) is chosen sufficiently large so that the inequality for a = 1 is automatically satisfied. Remark: We shall take up a higher dimensional analogue of the equation (5.4.31) again in Chapter 6, where it is easily solved via minimization techniques. Moreover, other examples of elliptic boundary value problems amenable to (5.4.29) are given in Section 5.5E. We now proceed to the more general case of a mapping f = L + N of a Banach space X into a Banach space Y , where L is a linear Fredholm operator of indexp > 0.

(5.4.32) Theorem Let P denote the canonical projection of Y onto coker L, where L E @,,(X, Y ) as mentioned above. Then if N E M ( X , Y ) is compact and uniformly bounded and the following hypotheses are satisfied: (i) limR+m P N ( R a + x ) = ~ ( a #) a holds uniformly for x E (Ker L ) I and llxll uniformly bounded where a E Ker L with llall = 1; ( 4 I l P N ( R a + X l l l < Il?l(a)ll; , E L[ ~ ( a )#] 0 ; (iii) the stable homntopy class of ~ ( a )limk+m then a necessary and sufficient condition for the solvability of the equation Lu N u = f i s IlPfll < lIq(a)ll for all a. Moreover, the mapping L + N has open range in Y .

+

Proof: Suppose f E Range(L

P N ( u ) = Pf>

+ N ) , then for some u E X

so by (ii)

IIPfll

< IlV(a)ll.

Thus the condition stated in the above theorem is necessary. On the other hand, if this latter condition hoids, we note that by virtue of (5.3.21) it

280


suffices to prove that the stable homotopy class of P[N(Ra)- f] # 0 (for R sufficiently large). But the conditions (i), (ii) imply that the two maps f , = P[N(Ra)- f] and f;, = P[N(Ra)]are homotopic on a large enough sphere (i.e., for R sufficiently large) via the simple homotopy tfi + (1 - t ) f2. Thus since the stable homotopy class of P[N(Ra)]# 0,by virtue of the hypothesis (iii) above, the stable homotopy class of P(N(Ra)- f) # 0 and the first part of theorem is proved by applying (5.3.21). The proof that L N has open range now follows as in (5.4.29) above.

+

An interesting consequence of (5.4.29) is the finiteness of the number of solutions of the equation Lu + Nu = f for almost all f in the range of L + N in the case index L = 0 and (provided N is sufficiently smooth), a corresponding result holds in case index L > 0. In fact we prove (5.4.32‘) Theorem Suppose the hypotheses of (5.4.32) hold and in addition X is reflexive while N is compact, then for any regular value f E Range(L N ) (i.e., by (3.1.45) apart from a possible set of first Baire category) the number of solutions of Lu Nu = f is either finite if index L = 0 or a compact submanifold of Y of dimension p if index L = p > 0, and N is sufficiently smooth.

+

+

+

Proof: We first suppose f E Range(L N ) and index L = 0, and we shall prove that (5.4.32) implies the fact: (*) any sequence {u,}such that 11 Lu, + Nu, - f l l + 0 is uniformly bounded in norm. Assuming the validity (*) for the moment, we note that i f f is a regular value of L + N the solutions i? of Lu N u = f are isolated (since L N ’ is invertible), while (*) implies that if the solutions { i?} of Lu Nu = f a r e infinite, they are certainly uniformly bounded. In that case the weak compactness of bounded sets in a reflexive Banach space and the compactness of N imply for some weakly subsequence {u,} that Lu, is strongly convergent. Consequently by the property (1.3.27) of Fredholm operators { u , } is strongly convergent. This contradicts the fact that the J solutions of Lu + Nu = f are isolated. Consequently, these solutions must be finite in number. The case for index L > 0 follows from the remark after Smale’s theorem (3.1.29). Finally, we demonstrate the fact (*), and to this end we suppose the sequence u, satisfies the condition that F(u,) = Lu, + Nu,,- f tend to zero in norm. Then, decomposing u,, in the form u, = o, + z, with z, E Ker L and v, E X , , we find

+

IlL%ll

Q

IIF(ui7)ll +

+

+

IlNu, -fll*

Thus, the uniform boundedness of N implies the uniform boundedness of 11 LU,)~.On the other hand, since L is invertible off Ker L, { I ( o,ll> is also

5.4

HOMOTOPY A N I ) M A P P I N G PROPERTIES OF N O N L I N E A R OPERATORS

28 I

uniformly bounded. Thus to verify (*) it suffices to show that { I I z n l l } is uniformly bounded. But this follows by supposing otherwise and obtaining a contradiction. For then, from the fact that f E Range(L + N), (5.4.29) implies that if I(z,II+ 00, IIPofll < IIP,N(z,, + u,,)ll: but since F(u,) +O, P,(N(z, + c,,) - f ) + O , so that we have the desired contradiction. 5.4F

Properties of cone preserving operators

A real Banach space X admits a cone K if K is a closed convex subset of X such that (i) x E K implies ax E K for any nonnegative real number a , and (ii) x E K implies - x g K , unless x = 0. Many of the results proved in earlier subsections can be considerably sharpened for mappings f that map a cone K of X into itself. Such mappings are called cone preserving. As an example of such a sharpening, we prove the following extension of (5.4.24).

(5.4.33) Theorem Let D be a bounded open domain (containing the origin) in a Banach space X that admits a cone K . Suppose N is a compact cone preserving mapping such that (5.4.34)

IlNxll

>0

Then the equation x n K).

for x = ANx

E(

K n aD).

has a solution (x,, A,) such that A,

> 0 and

x, E (aD

Proof: We argue as in (5.4.24). by reducing the problem to a finitedimensional one. Indeed, if x = ANx has no solutions as described in the theorem, then by (5.4.34) and the compactness of N , there is a number a > 0 such that

(5.4.35)

llNx - t x l ( 2 a

>0

for

t

>0

and

x E aD

n

K.

Then by (2.4.2), there is a compact operator N , with odd finitedimensional range X , such that 11 N,x - Nxll < f a for x E aD n K . Consequently, by (5.4.34) for i > 0 and any x E aD n K , IIN,x - txll > : a ; so that the equation x = AN,,x has no solutions (x, A) with x E aD n K and A > 0. We shall contradict this last statement by providing the theorem for the case when X is an odd finite-dimensional Banach space; for then the equation x = AN,x will have a solution (x, A) with x E X , n ( aD n K ) and A > 0. In case X has a," odd finite dimension, s_uppose for the moment that N, has an extension N onto aD such that (i) N maps aD onto K and (ii) /I N-xll > 0 for x E aD. Then by the argument of (5.4.24), the equation x ANx has a solution (x,, A,) with x, E do. We shall prove that d,(Z A N , 0, D) = 0 for A sufficiently large, so that A, > 0. Once this fact is

282

5


-

I

established, x , = A,Nx, E aD n K so that N x , = Nx,; whence the equation x = XNx has the solu_tion (x,, A,) with x, E aD n K and A, > 0. To prove that d,(Z ; AN, 0, D ) = 0 for A sufficiently large, we show that the vector field x - ANx omits a direction for A sufficiently large. Indeed, let u E K ( u E 0) and suppose that as A,, + 00 there is a sequence ( t , , x n ) such that (5.4.36) x, - h,Nx, = t,,u with x, E aD and t,, > 0. Then (after possibly passing to subsequences), we may suppose that N x , + v ; and since inf llNxll > 0 for x E aD, v # 0 and D E K . Consequently x , / h , - N x , .+ - v , but by (5.4.36), xn/An - N x , = t,,/A,u. This is a contradiction since - v $? K , while t,u/A,, E K for all n. Finally, we show that the extension i exists, where N ( a D n K ) lies in X,, a k-dimensional linear subspace of X and IINxll 2 p > 0 for x E ( a D n K ) . Then i3 N ( 3 D n K ) does not contain 0. Furthermore, by choosing an orthonormal basis in X , , we may write I

for x E a D n K , N ( x ) = ( n , ( x ) ,. . . , n k ( x ) ) and by Urysohn’s lemma, we may suppose that the functions n , ( x ) are extended with preservation of continuity to functions ii,.(x) on 0. Now to construct N , let y E int co N ( a D n K ) and let r ( x ) be the continuous retraction of X , onto co N ( a D n K ) defined for x !Z co N ( a D n K) by constructing the line segment L ( x , y ) joining x and y in X , and letting r ( x ) be _the point of intersection L ( x , y )n a i 3 N ( a D n K ) . Now we define N x on B, by setting N x = Pn”(x), where Z ( x ) = ( f i , ( i ) , . . . , n”,(x)). Clearly for x E aD n K , i,= N x , so that i is a continuous extension of N to 5. In addition, N ( 5 ) c i3 N ( a D n K ) c K and for x E 0, llixll

Thus

> d(i3

N ( a D n K ) , 0 ) > 0.

i is the desired extension.

An important consequence of (5.4.33) is the following important theorem of Krasnoselski (1964). (5.4.37) Theorem on Monotone Minorants Suppose N is a compact cone preserving operator defined on the cone K , such that there is a linear cone preserving operator L (i.e., a monotone operator in the sense that it preserves the order relation in K ) and a nonzero x , E K such that (5.4.38)

Nx

> Lx

and

Lx,

> ax,,

where a

> 0.

Then for any bounded domain D (containing 0), the equation x solutions ( x , h) with x E a D n K and h > 0. Proof: Let N , x = N x

+ exo for any E > 0. Then N,

= ANx

has

is compact, and for

5.5 A P P L I C A T I O ~ STO N O N L I N E A R BOUNDARY V A L U E PROBLEMS

283

x E aD n K , 11 N,x\l 2 inf,,>,x, lIy(1> 0. Hence by (5.4.33), there is a pair (A,, x,) satisfying

(5.4.39)

Nx,

+ cxO= A,x,,

with

A,

>0

and

x,

€aD n K .

Now as E + O (after possibly passing to subsequences), we may suppose that Nx, -+y and A, +A,. Clearly y E aD n K ; so that it remains only to prove that A, # 0. To this end, we first note that (5.4.38) implies (5.4.40)

Lx,

+ exO < A,x,

and

x, 2 A,p'cxo.

Thus, there is a largest number i, > 0 such that x , > t,xo, implying that Lx, > t,axo. But (5.4.39) implies that L x , < A,x,. Therefore, x, > t,aA,-'xo, and by the maximality of i,, 1, 2 i,a/A, so that A, 2 a > 0. Consequently, A, > 0, as required. 5.5 Applications to Nonlinear Boundary Value Problems

The results of Sections 5.3 and 5.4 are of immense value in proving qualitative results on the structure of the solutions of boundary value problems for nonlinear partial and ordinary differential equations. In particular, questions that we shall consider here include (a) existence (or nonexistence), (b) uniqueness (or nonuniqueness), (c) continuous dependence on a parameter. as well as (d) continuation of solutions of problems depending on a parameter. In general, the following steps are necessary to apply the abstract results proven in the previous sections to concrete problems. First, any parameters implicit in the nonlinear systems should be introduced explicitly by appropriate coordinate transformations. Secondly, suitable Banach spaces X and Y must be chosen so that the differential system under consideration can be represented as a well-defined mapping f defined on a domain in X with range in Y . Next one must prove the basic boundedness, continuity, and differentiability properties of f that are necessary to apply the appropriate degree theory to the problem at hand. Finally, one must prove the analytical estimates necessary to calculate the degree off. We begin by considering an analogue of the original problem discussed by Leray and Schauder in their fundamental paper (1934). 5.5A

The Dirichlet problem for quasilinear elliptic equations

Let i2 be a bounded domain in R N with boundary following system of equations defined on

a,

aQ, and consider the

284


(5.5.1)

2

(5.5.2)

l4+I PI = 2 u l a a = g.

+

A , ~ ( X , u , D U ) D " D ~ A,(x, u,

DU)= o in 3,

The classic Dirichlet problem for (5.5.2)-(5.5.3) consists in determining a function u E [ C L ( 0 )n C(a)] that satisfies (5.5.1)-(5.5.2) in the pointwise sense. The differential operator on the left-hand side of (5.5.2) is elliptic if there is a constant p > 0 such that A a p ( x , y , z)t& 2 pIt12 for Iyl, Izl Q M, x E The examples given in Section 1.2 show that such quasilinear elliptic Dirchlet problems may not be solvable for a variety of reasons, including the shape and size of D or the rapidity of growth and sign of A,(.% y , z ) . The question of the solvability of Dirichlet problems of this class was posed by Hilbert in 1900 in his famous address (Hilbert, 1900), and was studied extensively by S. Bernstein thereafter. In N = 2, Bernstein attempted to solve (5.5.1)-(5.5.2) by (a) introducing a parameter t explicitly into the system (5.5.1)-(5.5.2) to obtain a one-parameter family of systems P, so that for t = 0, the system Po is solvable, while for t = 1, the system P I coincides with (5.5.1)-(5.5.2); and (b) showing that each P, is solvable for t E (0, I], by continuation. The continuation method was greatly extended in 1934 by Leray and Schauder, who transformed it into a homotopy argument by means of the degree. This approach, however, requires difficult analytic a priori estimates in order to ensure that the degree of the mappingf can be defined. Once these estimates have been established, the basic idea is to apply the a priori bound principle discussed in (5.4.14). As a simple example, we mention

a.

(5.5.3) Theorem Suppose a 3 and g are of class C3, while the functions A a p ( x ,y , z ) and A,(x, y , z ) are C in x, y , z . Then the Dirichlet problem

'

for (5.5.1)-(5.5.2) is solvable provided any solution u1 of the system obtained from (5.5.1) by replacing A , by tA, and g by rg for t E [0, 11 satisfies the a priori estimates

where M , and M , are constants independent of t and u,. Proof Sketch: As mentioned above, we apply (5.4.14) to prove the result, but first it is necessary to determine an appropriate Banach space X for the operator. To this end, we follow the Schauder inversion method discussed in Section 2.2D. Indeed, the a priori estimates (5.5.4) show that any solution u1 of a member of the adjusted system has a Holder continuous gradient with exponent a E (0, 1) and independent of t and u,. We let X = C 1 9 a ( a and ) , define a mapping T : C ' * a ( a into ) itself by fixing u E X

5.5

285

APPLICATIONS TO NONLINEAR BOUNDARY VALUE PROBLEMS

and considering the solution U of the linear elliptic Dirichlet problem A , ~ ( X ,u.

+ ~ , , ( x ,u , D U )= o

DU)D"D%

on

3,

u l a Q= g.

l"l+ IPl=2

By the results of Section 2.2D, Tu = U maps X into itself, is bounded, and in fact, maps bounded sets of X into bounded sets of C 2 . " @ ) . Since C 2. "(D) is a compact subset of C the mapping T is compact. Now we apply the a priori bound principle (5.4.14) with f(u, f ) = tTu and X = C ' . a ( G ) .By hypothesis, if L' satisfies u = f T u , then L' E C2."(Q), and also satisfies the adjusted system. Consequently, Iullcl(~)< M , M,. Furthermore, by the a priori estimate (5.5.4)mentioned at the beginning of the proof, and a Holder continuity of Ladyhenskaya and Uralsteva (1968), there is a number a E (0, I),

"(a),

+

I V ~ ( x -) Vu(y)l where M , is independent of Ilollcl.qn,

Q

f

Q M,lx

-yl",

E [0, 11 and u. Thus

MI + M , + M ,

implying that, by (5.4.14), (5.5.1)-(5.5.2) has a solution u E C 2 ( Q ) .See Ladyhenskaya and Uralsteva (1968) for more details. 5.5B Positive solutions for the Dirichlet problem for Au f ( x 7 u ) = 0

+

An interesting application of the Schauder fixed point theorem is concerned with the positive solutions of the following Dirichlet problem defined on a bounded domain 3 c R N , (5.5.5) A u + h ' f ( x , u ) = 0. f ( x , u ) > p > 0 for u > 0, u I = 0. We prove the following extension of the result established for system (1.2.3)-( 1.2.4): (5.5.6) Suppose in addition to the fact thatf(x, u ) > ,i3 > 0 for u > 0, that f ( x ? u ) is nondecreasing in u for fixed x and thatf(x, u ) > g ( x ) u for u > 0. Then there is a finite (critical) number A, > 0 such that for X < A, (5.5.5) has at least one positive solution, while for h > A, (5.5.5) has no positive solution. Proof: The argument can be divided in a natural way into three parts. First it is shown that under the given hypotheses (5.5.5) has a solution for some h > 0. Next it is shown that if (5.5.5) has a positive solution for A, > 0, then it has a positive solution for all h in the interval (0, A,], Finally, we show that for all h sufficiently large, (5.5.5) has no positive solution. (i) (5.5.5) has a positioe s o h i o n for .Tome A: Observe that the positive solutions of (5.5.5) are in one-to-one correspondence with the positive solutions of the integral equation (5.5.7)

u = h * / n C ( x . y ) f ( x ,u).

286


Now observing that Green’s function C ( x , y ) > 0 in and is integrable over 0, we conclude that for u 2 0 ( 5 . 5 .8 )

TU =

J,~ ( x , y ) f ( x u, ) > ~ & ( x , y ) = y

(say).

Let C(0) denote the Banach space of continuous functions defined on 0 with the sup norm, then the operator Tu just defined is clearly a continuous and compact mapping of the positive cone of C(P) into itself. The inequality (5.5.8) ensures that the same statement can be made for the mapping Su = T U / ~ ~ TIn Ufact, ~ S \ ~is a~ continuous ~ ~ compact mapping from the bounded closed convex set C + = ( u I u a 0, llullc G I ) into aZ+ = { u I u > 0, llull = 1 and a fixed point ii of S is a solution of ( 5 . 5 . 5 ) with A* = 1/11 Tiill. Now the Schauder fixed point theorem (2.4.3) implies that S has a fixed point ii €ax+,and therefore (5.5.5) has a positive solution with X > 0. (ii) If (5.5.5) has a positive solution u,, for A, > 0, then (5.5.5) has apositive solution for all h in the interval (0, A,]: Let TAu= X2jQG(x,y ) f ( x , u). Then for X E (0, A,], we shall show that Th maps the closed convex bounded set Z, = ( u I 0 G u G u,, u E C(0)) into itself. Since T, is continuous and compact, Schauder’s fixed point theorem will again imply that T, has a fixed point u,, in 2 , and U, will satisfy (5.5.5). To show that TA maps Zo into itself, we note that since f ( x , u ) is nondecreasing in u and G ( x , y ) > 0 in 0, for u E Z, f(x, 0)

f(x9

u) G

f(x9

uo)

and JnGf(x. 0) &f(x.

G &f(..

uo),

Thus for h E (0, A,] and u E C,, 0 < T,(O) G T,(u) G T,,(u,) = u,, as Th(u)E X,, as required. (iii) For X sufficiently large, (5.5.5) has no positive solurion: If (uI, XI2) denotes the first eigenfunction and eigenvalue of Au + X2g(x)u = 0 subject to the Dirichlet boundary condition u I I an = 0, then u I > 0 in 0. Thus multiplying ( 5.5.5) by uI and integrating by parts twice, we find that if u satisfies (5.5.5), then

0=

>

/ ( A u + h2f(x, u ) } u I =/a( - A i g ( x ) u l u cl

+ h2f(x, u)}

1-s

(A* - h:)g(x)uIu.

Thus we arrive at a contradiction of h

> A,

5.5C Periodic water waves Here we consider the classic problem of proving the existence of steady periodic waves at the free surface ar of an ideal incompressible fluid, under gravity. Because of their precision and relative simplicity, the results described here represent one of the most successful attempts to apply our analysis to a given difficult nonlinear eigenvalue problem. We suppose the flow is steady, irrotational, and two dimensional, the fluid occupying a domain r in R2. The points in R2 are denoted by Cartesian coordinates ( x , y ) . Euler’s equation of motion and the equation of continuity for this problem then become

(5.5.9)

Al= 0

in

(5.5.10)

;)V{1*+gy=const.

on

r, ar,

287

5.5 APPLICATIONS TO NONLINEAR BOUNDARY VALUE PROBLEMS

FIG. 5.3 Notation for periodic water wave problem.

where 1denotes the velocity potential for the flow. Hence we are forced to solve a nonlinear free boundary value problem. Following an argument due to Levi-Civita, one introduces the complex variable z = x + and two analytic functions of z ,

+

u ( z ) = { + i#

and w=log

Here 4 is the stream function for 5, @ is the angle formed by the velocity vector I/ at the point ( x , y ) , and C(@)is the harmonic conjugate of @. In order to work in a known domain, one chooses u = 1 i+ as an independent variable and regards o as a function of u. Assuming, for simplicity, that that the fluid is at infinite depth, and after performing the recommended period transformation, the desired periodic solutions are in one-toone correspondence with the nontrivial solution of the nonlinear integral equation

+

(5.5.11)

@(B)

= h / T K ( B ’ , B)e3C(@) sin @

dB’,

0

where X = ( gu)/2ncZ, G is the wavelength, and c denotes the constant horizontal velocity of the moving wave. K(B’, 0) is the Green function associated with the Neumann problem for A in a circle, and the additive constant in the definition of C(@)is so chosen that 12 C(@(O))dB = 0. Note that (5.5.11) is in the form of a nonlinear eigenvalue problem. See Fig. 5.3 for notation. There are basically two types of problems associated with (5.5.1 1): (i) a local bifurcation problem for @ very small, and (ii) a general global problem for l@l unrestricted. The local problem was “solved” in 1925 by Levi-Civita, but the global problem (which we discuss here) remained only

288


O

X

FIG. 5.4 Limiting form at the crest of a periodic water wave.

partially solved until 1961 when the Russian mathematician Y. P. Krasovskii proved the following results. (5.5.12) Theorem There exist steady periodic wave satisfying (5.5.9) and (5.5.10) for which the maximum angle of inclination of the tangent to the wave profile takes any value in the open interval (0, n/6). The wave is symmetric relative to a vertical axis passing through the peak of the wave. Furthermore, waves of this type with arbitrarily large Froude number A cannot exist. Before sketching the proof of this interesting result, we note that the number n / 6 appearing in the theorem is sharp in the sense that (i) Stokes’ periodic “limit” waves have max I@[ = n / 6 (see Fig. 5.4) and possess cusps, (ii) the solutions of (5.5.9), (5.5.10) show that steady periodic waves with max I@I > n / 6 do not exist (see Wehausen (1969) for further information). Actually, Krasovskii proved a sharp analogue of Theorem 5.5.12 for waves of finite depth and periodic bottom by slightly modifying the proof given below.

Sketch of the proof of Theorem (5.5.12): the following steps:

The proof breaks down into

(1) representation of the equation (5.5.11) as an operator equation of the form x = A A x in a suitable Banach space X ; (2) proof of complete continuity of the map A in X ; (3) application of the Leray-Schauder degree to the operator equation: (4) proof of the estimates necessary to calculate the Leray-Schauder degree.

In order to carry out the steps (1)-(4), we need to know the following

APPLICATIONS TO N O N L I N E A R B O U N D A R Y V A L U E PROBLEMS

5.5

289

analytic facts concerning the conjugation operator C of a harmonic function and the kernel K ( 8 ’ , 6). (Interestingly, the limiting number a / 6 arises naturally from these facts and the requirement that the operator A be completely continuous.)

Lp estimates for the boundary values of conjugate harmonic functlons Let u ( z ) be a harmonic function defined in the unit circle IzI < 1 of the complex plane. with boundary values u(eis) E Lp[O,2771 ( I < p < 00). Then, if v ( z ) denotes the harmonic function conjugate to u ( z ) in IzI < 1 and normalized by setting J$’ t ( f ) d6 = 0, f(z) = u ( z ) + i u ( z ) is analytic in It1 < 1, and f(0) is real. Now we define the linear mapping C ( u ( e ” ) ) = u(e”), and inquire about Lp boundedness of C. In this connection, we have the following results: Fact 1 Theorem of M. Riesz For 1 < p < co, C is a bounded map of Lp[O,2771 + Lp[O, 2a], and thus there is a constant cp independent of u such that IICuIILp < cpllull$.

Fact 2 Zygmund’s Theorem

If IuI Q I , then

These results can again be proven by the method of singular integral operators, by extending (1.3.1 8) “to the periodic case,” see Zygmund, (1934). Fact 3 Lp estimates for K maxb j;lK(6’, 6 ) l p do’ < Cp. and for 1 < p < 00 and fixed 6. a K / a O maps Lp[O,2a]+ Lp[O, 271 boundedly. This third fact is a well-known property of Green’s function for A. Steps 1 and 2: Now let X = C,[O, a],i.e., the continuous functions on [0, a] that vanish at 0 and a. Let l i @ l l x = sup[,,1 I@(S)l and define the operator

(5.5.13)

A @ ( B ) = j T K , ( H ’ ,B)e’C‘o’sin 0 d6”. 0

One shows that A is a completely continuous map defined on the sphere S(0, p ) of radius p < a / 6 in X. Note that a / 6 comes up naturally by combining (5.5.13) with Fact 2:Clearly, by Facts 1 and 3 above, A is a well-defined and continuous map from S(0, p ) + X for p < a / 6 . In fact, under Holder’s inequality, one easily shows that for @,, Q2 E S(0, p), p =~ / 6 d ( d > 0).

I

w.

I - ~~ 4 ~Q1K1~ I I Q -,

290


To verify the compactness of A , we again-use Facts 1 and 3 to show that if @(8) = A @ , then for some s > 1, Ild@/d8-(lL,< K d , s for @ E S(0, p ) with p = 7 / 6 - d (as above). Consequently, ~ ~ @ ~ 0. The desired compactness of A thus follows. (Here do, is the Banach space of Holder continuous functions of exponent p.) Step 3: In order to apply the Leray-Schauder degree to prove the existence of a solution of (5.5.1 I), we let

I

I + L"

A ~ ,= ~ A@ A @

c

K ( 8 ' , 8) sin 8' d8' .

Note that A , is compact (and positive). We prove (t):the Leray-Schauder degree of Z - A,, Aon the positive cone K p = {@(8) I @ E Co[O, 71, @ 2 0, l)@llc0 < p ) , 0 < p < 7/6, is different for large and for small A. The last part of Theorem (5.5.12) and (t)suffice to prove the existence part of Theorem (5.5.12). To see this, we first note that (t)implies that there are sequences { A n } , ( e n ) , {@,} with A, > 0, en+O, and 0,E C,[O, 71 such that

Il@,ll

@, = A,A,@,?

c = P.

By the compactness of A and the boundedness of [A,\ (due to the nonexistence part of Theorem (5.5.12) there is a (strongly) convergent subsequence {A,) and {@,J with limits (Ab, QPp)such that (5.5.14)

aP= A , A @ ~ ,

Ipbllc= p,

Q p ( 8 ) 2 0 on

[0, 71.

Thus one can extend @(8) to an odd 27-periodic function of 8. Step 4: First we prove (t) mentioned in Step 3 above. For A very small, d(Z - A,,A, 0, K p ) = 1 since for h = 0, Ac, = 0. On the other hand, for A very large, @(8) - A c , , @ ( 8 ) cannot be positive for max IQ(8)l < p, so d(Z - A,,A, 0, K p ) = 0 in that case.

The nonexistence result of Theorem (5.5.12) is somewhat more difficult. It is based on the following two a priori estimates for solution @(a) of (5.5.1 1): (5.5.15) There are absolute positive constants y and 6 such that @.'(8) 2 ( A / 6 ) Y L ( @ y ) , where L@= / g K ( 8 ' , 8)@(8') do' provided ll@pllco < 7/2. (5.5.16) There is an absolute constant

p > 0 such that

OY(f?) 2 /3 sin 8.

Assuming (5.5.15) and (5.5.16) with P maximal, the proof of nonexistence is as follows. Applying the operator L to (5.5.16) and using (5.5.15), we have

(a

).'@y

2

m ( e ) > pL(sin e) = p sin 8,

5.5

A P P L I C A T I O N STO N O N L I N E A R B O U N D A R Y V A L U E PROBLEMS

291

i.e., @ y > ( X / C ? ) ~sin ~ 8. so that (X/S)y < 1. Hence for X > 6, (5.5.1 1) can have no solution. To end our sketch of the proof of Theorem (5.5.12), we prove (5.5.15) and (5.5.16). To demonstrate (5.5.15), it suffices to show that for @ E K p (5.5.17)

L(e3C'0)sin a) > 1 L(@y)l'y.

Now (5.5.17) follows from Holder's (inverse) inequality since

> L ( ~ ~ c ( * ) Q> )

~ ( e 3 ~ (sin *)

with I / p + l / y = 1 ( q < O ) . Then the basic Facts 2 and 3 imply that for - 1/10 and [@I < 7r/2. IL(e31q1C('))I < 6 -Iq1. Finally, we prove (5.5.16). Applying the inequality (5.5.15) k times and letting (Py(0)= C,"=lansin no, we find

q=

oc

(5.5.18)

L k Q y=

2 n=I

an k

sin n0

n

Furthermore C ( a n / n k )sin n0

al/2. 5.5D The continuation of periodic motions of autonomous systems

We consider the periodic solutions of the second-order system (5.5.19)

x + A x + f ( x ) = 0;

If(x)l = o(Ix1).

Here x ( t ) is an N-vector functim of t , A is an N x N positive definite matrix, andf(x) is an odd, C 2 , N-vector function of higher order in x . In Section 4.1, we investigated the periodic solution ot (5.5.19) near the singular point x = 0 by means of bifurcation theory. Here we shall focus attention on the global structure of the periodic solutions of (5.5.19). As a first result, we consider the global analogue of Liapunov's theorem (4.1.4).

292


(5.5.20) Theorem Suppose the positive eigenvalues of the nonsingular matrix A , A:, A;, . . . , A;, are such that for some integer j (1 < j < N ) , (5.5.21)

Ai/Aj # integer

for i

=

1,. . . ,N

(i # j).

Then (5.5.19) has a family of periodic solutions X(E) of period depending continuously on a real parameter E, and such that

T(E)

(i) as 6-0, x(E)+O and 7(~)+27r/A,; while T ( E ) + ~ or x ( c ) + O and T ( E ) (ii) as E + co, either sup IX(E)~ +2sn/Ak for k = 1 , . . . , N and n = 1 , 2 , . . . ; but if n = 1, then k # j (i.e., as either the amplitude of x(E)+ to, the period T ( E ) + to, or X(E) tends to a covering (possibly multiple) of the periodic solution of the linearized equation).

+

Proof: By repeating the argument of Section 4.1 and setting t = ;Is, the odd periodic solutions of (5.5.19) are in one-to-one correspondence with the solutions of the operator equation

(5.5.22)

x = A'{ 4 x

+ %(x))

in the Sobolev space H = 6'1,2{[0,7r]; R N } .Here the operators 4 and % are completely continuous and defined implicitly for x,y E H by the formulas (5.5.23)

( s x , y ) = / n A x ( s ) * y ( s ) ds; ( % x , y ) = / y ( x ( s ) ) y ( s ) ds. 0

0

The condition (5.5.21) implies that the eigenvalue A,? of 4 is simple on an appropriately chosen closed subspace of H (see Section 4.1 .C). (Cf. Section 4.1.) Consequently, by Theorem (5.4.28), there is a continuum of solutions of (5.5.22), ( x ( E ) ,T ( c ) ) , joining (0, l/Aj) either to co or to (0, N */A;), where N = 1, 2 , . . . and k = 1 , . . . , N , with k # j in case N = 1. Thus the theorem is proved. We sharpen the result just obtained by imposing other restrictions on the vector function f ( x ) in (5.5.19). One important class of results can be obtained by using the result (5.4.37) on monotone minorants. Suppose, for example, that we set g(x) = A x + f ( x ) a n d write g(x) = (g,(x), . . . , g N ( x ) ) , where x = (xl, . . . , x N ) . Then we prove (5.5.24) Theorem Suppose g(x) is an odd function of x with the properties: (i) g i ( x ) Z 0, whenever xi Z 0 ( i = 1, . . . , N ) ; and (ii) there is a constant k > 0 and an integer j (1 < j ~ ( x 2) kxj for all nonnegative vectors x.

0. Consequently. by extending these solutions to ( - M, a),as odd periodic functions of s, this family corresponds to the desired family of periodic solutions of (5.5.19). 5.5E Necessary and sufficient conditions for the solvability of coercive semilinear elliptic boundary value problems

We begin by considering the following semilinear Dirichlet problem defined on a bounded domain D c R N : et4

(5.5.29)

+f ( u ) = g

Dau

I

= 0,

la1

< m - 1,

294 where


C

is the formally self-adjoint operator fu =

2

14. I Bl

( - 1)'"' D a { U a , ] ( X ) D 4 d } ) , m

and the function f satisfies the following hypothesis: (*)

< 03 < f(s) < f ( + 03).

lim f(s) = f( 5 03) S+?

m

exist and moreover f( - 03) We prove the following result:

(5.5.30) Theorem A necessary and sufficient condition for the solvability of (5.5.29) is for each zo of L2(Q)of norm 1 in Ker f the following inequalities hold

Proof: Using the duality method Section 2.2D of representing (5.5.29), we obtain a Hilbert space reformulation of (5.5.29) as the operation equation

(5.5.31)

Lu

+ Nu = g

with L and N mapping the Sobolev space W,,,*(Q)into itself defined implicitly by the formulas

Now our result (5.4.29) applies directly to (5.5.31) since, as we shall see, the hypothesis (*) implies that the condition (5.4.30) is automatically satisfied. In fact we show that

uniformly for c l Ker L and Ilt1\1 uniformly bounded. Once we demonstrate (5.5.32), our result will follow directly from the result (5.4.29). To this end we denote the right hand side of (5.5.3 I ) by n(zo)and note that by definition ( N ( R z o + o), z o ) = j f ( R z o Q

Thus given (5.5.33)

E

+ u)zo.

> 0, we shall show that

ln(zo) - j f ( R i o + u)zo dxl< n

E

5.5 APPLICATIONS TO N O N L I N E A R B O U N D A R Y V A L U E PROBLEMS

for R sufficiently large. First, we note that there is a S

> 0, such

295

that

for any measurable set A , with m ( A ) < S. m ( A ) here denotes the Lebesgue measure of the set A . We note that S can be chosen independent of zo E Ker L of L, norm I since Ker L is a finite-dimensional space. Now, for any 2: with IIt‘ll < k . letting 3, = ( x E 3, Io(x)l < N ) . we can choose N sufficiently large so that m(3 - 3,) < 8 for all such u. Thus the left-hand side of (5.5.33) is less than or equal to

The last two terms of the above are less than ~ / 4for the reasons just stated. Next we show that for R sufficiently large, the first two terms are each < r/4.This follows by the Lebesgue convergence theorem since for lo1 < N , on Q + n3,, the integrand in each tends to zero pointwise and is bounded by an integrable function. Thus we have shown that (5.5.33) holds, and we have completed the proof. The result (5.5.30) concerning the Dirichlet problem (5.5.29) can be substantially extended by utilizing (5.3.22) (5.4.32). Indeed let Pu denote an mth order elliptic system of equation k unknowns with “coercive” boundary conditions BU expressed in terms of differential operators of order less than m. Then let f(x. D a u ) denote a continuous bounded vector-valued function of the vector u such that limR-,mf(x, Ru,) exists uniformly. Then on the basis of (5.4.32) we can find necessary and sufficient conditions for the solvability of the elliptic system Pu

+ f(x, Dl‘u) = 0

on

Q

c w[,

Bu I a a = 0. Such a system can be represented by an operator equation with domain a Banach space X of vector-valued functions each component of which is an element of the Sobolev space W,,,p(3)and satisfies the boundary conditions B. The range of this mapping is an L, space of vector-valued functions. Moreover the abstract associated linear operator L defined by (P,B ) defined on such a Banach space X is well known to be a Fredholm operator with discrete spectrum, so that the hypothesis (5.3.23) of (5.3.22) is generally satisfied.

296


NOTES A Further llnearlzatlon results lor proper nonllnear Fredholm operators It is natural to attempt to extend the results of Section 5.1 on linearization to a more general context. In this connection, the following result related to the Banach-Mazur theorem holds: (I) Let f be a nonlinear Fredholm operator of index p > 0 acting between Banach spaces X and Y. Then if f is a proper mapping, f must possess singular points (cf. Berger and Plastock, 1977). The proof of this result is based on the observation that if the mapping f possessed no singular points, f would necessarily determine a “fibration” between the spaces X and Y (see Spanier, 1966). Then by a corollary of the covering homotopy theorem the contractibility of Y would imply the contractibility of f-’(y) for each y E Y, which would contradict the fact that f- ‘ ( y )is a compact orientable p-dimensional manifold. B Addltlonal results on mapplngs with slngular polnts

The arguments used in the proof of (5.1.14) to determine the structure of the range of the operator A defined over a bounded domain 52 of R N by A ~ = A u + ! ( ~ ) , u/an=o, can be generalized in various directions. Among these are the results of Podolak (1976):

(I) Suppose the asymptotic condition (5.1.13’) is replaced by (5.1.W) A,-’ < lim f ’ ( r ) < A, < lim f ’ ( t ) < A,+,,

,

14-m

t++m

where A,- I , A,, A,+ denote three successive eigenvalues of A. Then a result analogous to (5.1.14~)holds, provided we suppose dim Ker(A + A,) = 1

and

kkIukI

# 0,

where u, is an eigenfunction associated with A,. In particular, g E 0, in (5.1.14~)means that the boundary value problem

AU + f(u) = g, ~ l a= n 0, has at least two solutions. (2) More generally, suppose L E Q p ( X , Y) is a linear Fredholm operator of index p a 0 with dim coker L = 1 and N is a compact nonlinear mapping of X into Y, satisfying a global Lipschitz condition (with a sufficiently small Lipschitz constant) and the asymptotic condition lim N ( t u ) = n(u), 1-m

t

Then if Po denotes a projection of Y onto coker L and Pon(xo) # 0 for all elements x,, E Ker L of norm I, a result analogous to (5.1.14~)also holds provided we interpret g E 0, as meaning that the equation Lu + Nu = g has multiple solutions (in the present case a compact submanifold of dimensions p ) . C Further properties and appllcatlons of the Leray-Schauder degree

(I)

Let D be a bounded domain of a Banach space X and suppose f and g denote

NOTES

297

compact perturbations of the identity. Then the following composition theorem holds for the Leray-Schauder degree of fg

for any 6, E A,, where A, denotes the bounded components of X - g ( 8 D ) . As an application of this result the following generalization of the Jordon separation theorem can be established. (2) Let D and D ’ be bounded open sets of a Banach space X such that there is a homeomorphism (i.e., a compact perturbation of the identity) between D and 0’.then the number of components of X - D and X - D ’are equal.

D Further results on the Dlrlchlet problem lor quasillnear elllptlc partial dlflerentlal equations Let P be a bounded domain in R N with smooth bonndary ail. Then we consider the solvability of the following quasilinear elliptic boundary value problem

2

a , ( x , u, V u )

l,J=I

(ii)

ti

a 2u = B ( x , u, V u) , ax,

ax,

1 an = 0.

Heref is a given continuous function, and we seek a smooth function u satisfying (i) at points of 0, and assuming the boundary condition (ii). Examples of geometric problems giving rise to (i)-(ii) include finding a nonparametric minimal surface or more generally a surface of prescribed mean curvature. Then the Leray-Schauder degree together with some rather subtle a priori bounds yields existence theorems for (i)-(ii). When applied to the equation

defined a hypersurface of constant mean curvature k over a smoothly bounded domain Q the following result is obtained: Theorem The Dirichlet problem in D for hypersurfaces of constant mean curvature is solvable for arbitrary C 2 boundary data if and only if the mean curvature H of the boundary surfaces satisfies the inequality H > [ n / ( n - I)]k for each point of the boundary. Moreover the solution is unique if it exists.

For a full discussion of such results we refer the reader to the paper by Serrin (1969). E Blbllographlc notes Section 5.1: The material discussed in this chapter section has an interesting history; see, for example, Hadamard (1904) and discussions of the monodromy theorem in complex variables texts. Our discussion of (5.1.1) is due to Plastock (1974). We also refer to papers of Browder (1954) and John (1968). The theory of operators satisfying conditions as in (5.1.6) are called strongly monotone operators and have been the subject of numerous recent papers and monographs. We refer the reader to Brezis (1968), Lions (1969), and Browder (1976). The reader will find the result (5.1.4) in the paper by Banach and Mazur (1934). Our discussion of Section 5.1B is based on papers of Berger and Podolak (1976, 1975). The nonlinear Dirichlet problem (5.1.13) together with the restrictions ( 5 . I , 13‘) was originally studied in Ambrosetti and Prodi (1972).

298


Section 5.2: Our discussion of this section is based on the work of Lions (1969) and Pohozaev (1 967). The Rayleigh-Ritz approximation for nonlinear eigenvalue problems are well discussed in Browder (1968) and Rabinowitz (1973). The result (5.2.29) on the steadystate solutions of the Navier-Stokes equations follows Fujita (1961), which in turn is based on papers of Hopf (1951) and Leray (1933). Section 5.3: The homotopy arguments in nonlinear problems of analysis began with Schauder. Our discussion of essential and inessential mappings follows Granas (1961). The Brouwer degree was extended to compact perturbations of the identity in the paper by Leray and Schauder (1934), where applications to the solvability of the Dirichlet problem for nonlinear elliptic equations can be found. Excellent treatments of the Leray-Schauder degree and its applications can be found in many books, including Krasnoselski (1964), Cronin (1964), Schwartz (1969), Nirenberg (1974). and Bers (1957), to mention only a few. The application of homotopy theory to compact perturbations of a linear Fredholm operator of positive index can be found in the papers of Svarc (1964) and Geba (1964). Applications of this theory to elliptic boundary value problems was given first by Nirenberg (1972); see also Berger and Podolak (1977), where the result (5.3.22) can be found. Our discussion of the generalized degree for proper Fredholm operators of index zero is adapted from Elworthy and Tromba (1970), and analogous results for operators of higher index can be found in Smale (1965); see also Palais (1967). Section 5.4: The result (5.4.1) can be found in Plastock (1974), while the result (5.4.5) is due to Krasnoselski (1964). The invariance of domain theorem is originally due to Schauder. The proof given here is adapted from Granas (1961). Rothe’s theorem (5.4.12) can be found in Rothe (1953). The result (5.4.24) can be found in Cronin (1973). The result (5.4.28) was originally proved in Rabinowitz (1973), the proof given here is due Ize (1975). The result (5.4.29) can be found in Berger and Podolak (1975), and its generalization (5.4.32) in Berger and Podolak (1977). The study of cone preserving mappings as discussed in Section 5.4F is based on arguments of Krasnoselski (1964) and has been the subject of a great deal of contemporary research; see Amann (1976) for a survey of recent work. Section 5.5: As mentioned in the text, the solvability of the Dirichlet problem for quasilinear elliptic equations was one of Hilbert’s problems in his address (Hilbert, 1900). An excellent recent survey with many new results can be found in Serrin (1969) and in the book by Ladyhenzskaya and Uralsteva (1968). The study of positive solutions of semilinear elliptic boundary value problems has been the focus of much recent research going far beyond the result (5.5.6); see Amann (1976), and Krasnoselski (1964) for good surveys. Levi-Civita’s early result on periodic water waves can be found in Levi-Civita (1925). A good survey from a physical point of view can be found in Wehausen (1968). Krasovskii’s paper containing the result (5.5.12) can be found in Krasovskii (1961). The study of continuation problems for periodic solutions of Hamiltonian systems dates back to Poincare. Interesting experimental results in this connection were obtained by Stromgren and his colleagues; see Stromgren (1932). The result (5.5.24) can be found in Krasnoselski (1964). The result (5.5.30) is due to Landesmann and Lazar (1970) and to Williams (1972). The proof given here is due to Podolak (1974).

CHAPTER 6

CRITICAL POINT THEORY FOR GRADIENT MAPPINGS

In this chapter we shall discuss some basic properties of operator equations involving gradient mappings. Since the zeros of a gradient mapping F’ are precisely the critical points of the real-valued functional F , we shall focus attention on those properties of F’ that can be discussed in terms of the geometry of the graph of F. The abstract results obtained have fundamental importance for rLLknyclassic problems that can be formulated in terms of the calculus of variations. In fact. we illustrate the application of our results by solving a number of problems in differential geometry and mathematical physics, as discussed in Chapter 1. The special properties of gradient operators and their associated critical point theory lend great insight to the general problems described in Chapter 5 . Here we shall see how these same problems can be studied by variational methods. These methods generally provide sharper information than can be obtained by more general techniques of Chapter 5. First, we study those critical points that can be obtained as the absolute minima of a functional F over a linear space X. Then we turn to isoperimetric problems, i.e., absolute ‘. lsoperimetric problems can be regarded as a simple minima of F over curved subspaces of A analytic method of studying saddle points of F when regarded as a functional on X. Finally. we discuss a more far reaching approach to saddle points, namely their classification and association with topological invariants related to the graph of F. The nonvanishing of these topological invariants ensures the existence of critical points of various types. Moreover, in each case we discuss applications of the results obtained to some pertinent problems of geometry and physics.

6.1 Minimization Problems

A fundamental heuristic principle of scientific understanding can be formulated as follows: “Many phenomena can be understood in terms of the minimization of an energy functional 4 ( u ) over an appropriate class of objects C.” Thus in Chapter 1, we described geodesics and minimal surfaces from this vantage. For the problems of mathematical physics, phase transitions, elastic instatility, and diffraction of light are amcng the phenomena that can be studied from this point of view. Indeed, the characterization of phenomena by variational principles has been a cornerstone in the transition from classical to contemporary phj 4cs. 299

300

6 CRITICAL POINT THEORY FOR GRADIENT MAPPINGS

Thus it is natural mathematically to study the simple and important class of critical points of a real-valued C I functional 4 ( u ) defined on an open set U of a Banach space X , namely the relative minima, i.e., points u o E U at which 4 ( u ) > 4 (uo) for all u near uo. Apart from the heuristic principle of the above paragraph, the importance of this class lies in the fact that such points not only are solutions of the gradient operator equation 4 '(u,)=O, but also possess remarkable stability properties. Roughly speaking, these stability properties are of two kinds: the first asserting that smooth perturbations of a functional 4 ( u ) possessing a relative minimum at uo must also possess a relative minimum nearby; the second asserting that oscillations starting near a strict relative minimum uo of 4 ( u ) always remain near uo. We remark that such stability properties are crucial both for the actual computation of relative minima and in the interpretation of their meaning in nature. In this and the next section, we take up the study of the minima of functionals and the applicability of our results to specific problems of general interest. 6.1A Attainment of infima

In a finite-dimensional Banach space X , any continuous functional defined on a closed bounded set M attains its infimum. As is well known, this property need not hold for infinite-dimensional spaces since closed bounded sets there are not necessarily compact. Thus in a Hilbert space H of infinite dimension, a bounded self-adjoint linear operator L with no point spectrum has the property that a = i n f ( l u , u ) over the unit sphere ax= { u llull = 1) is not attained on ax. Indeed, if a were attained on aS1, a would be an eigenvalue of L and hence in the point spectrum of L. The actual problem of determining restrictions on the closed subset M of a Banach space X and the functional 4 ( u ) to assure the attainment of the desired infima has been studied since Weierstrass first pointed out that such a functional may not achieve its infima. The basic restrictions involved center around the various notions of the compactness of the set M a = { u I u E M , 4 ( u ) < a ) and the lower semicontinuity of 4 . The concept of lower semicontinuity was introduced in our discussion of minimal surfaces in Chapter 1, and is a familiar property of Lebesgue integration (cf. Fatou's theorem). Weak lower semicontinuity of the functional 4 ( u ) is understood to mean that whenever u,+u weakly in X , 4 ( u ) < lim 4 (u,,). Thus, in accord with (1.3.1 1) the norm of a Banach space X is weakly lower semicontinuous on X . A simple result in this direction is

I

'

Historically the point proved to be crucial for the justification of Riemann's approach to potential theory (see the introductory chapter of Courant (1950)).

6.1

30 1

MINIMIZATION PROBLEMS

(6.1.1) Theorem Suppose 4 ( u ) is a bounded functional defined on a (sequentially) weakly closed and nonempty subset M of a reflexive Banach X . Then if 9 ( u ) is coercive on M (in the sense that 9. (u)+ co whenever 1 I u J 1 300 with u E M ) , and in addition 4 ( u ) is weak lower semicontinuous on M , then c = inf 4 ( u ) over M is finite and attained at a point uo E M . In particular, if M = X and 4 ( u ) is C ' , then 4'(uo) = 0 so that c = 4 (uo) is a critical value of 9. (u), and any element in 4 - ' ( c ) is a critical point of 4 (u). Proof: By the coerciveness of 4 on M , the set M a = ( u I u E M , 4 ( u ) < a } is bounded for any finite number a . Consequently, c = inf, 4 ( u ) is bounded above - co since the functional 4 ( u ) itself is bounded. Moreover, any minimizing sequence { u , } E Ma'+' is bounded and so has a weakly convergent subsequence (which we relabel {u,,)) with weak limit U. The weak lower semicontinuity of 4 ( u ) then implies that c = 4 (U) since

4 ( i i ) < lim !l(u,) = c

=

inf 4(u). M

Moreover, ii E M since M is weakly closed, and so ii = uo is the desired minimum. If M = X , and 4 ( u ) is C ' , then for any point u E $ - ' ( c ) , 4 ( u rh) > 4 ( u ) . Thus for any t E R 1 and h E X ,

+

(6.1.2) so that

( d / d t ) g ( u + rh)l,=,= ( 4 ' ( u ) , h ) = O ,

4 ' ( u ) = 0.

In order to investigate the applicability of this result, it is essential to derive criteria for a functional 4 ( u ) to be (i) weak lower semicontinuous, and (ii) coercive. The next two lemmas provide fairly general criteria, which will prove useful in the sequel. (6.1.3) Crlterlon for Weak Lower Semicontinuity A functional 4 ( u ) is weakly lower semicontinuous on a reflexive Banach space X if it can be represented as the sum 9 ( u ) = 4 , ( u ) + g2(u), where S,(u) is convex and g2(u) is sequentially weakly continuous (i.e., continuous with respect to weak convergence). More generally, 4 ( u ) is weak lower semicontinuous if 4 ( u ) = 4 ( u , u), where !I (x, y ) is a function defined on X X X with the properties that 4 (x, y ) is convex in x for fixed y , and sequentially weakly continuous in y uniformly over bou,nded sets in x E X . Proof: First we verify that a convex functional $(x) is weakly lower semicontinuous. By (2.5.2), if x, + x weakly in X , I

&(x,) - &(x) =

J', (x,

-

x, $'(x,(s)))

ds, where

I

=J', ( x , - x x ,

x,(s) = sx,

+ (1 - s)x

1

$ ' ( x ) ) d s + J (x,,-x, $'(x,(s))0

$'(x))ds.

302


By the convexity of &, the last integral on the right is nonnegative (since the integrand itself is nonnegative). On the other hand, since x, + x weakly in X , the first term tends to zero. Thus b [ & ( x , ) - &(x)] 2 0. Next suppose that 4 (x) satisfies the more general properties mentioned in the theorem. Then if x, + x weakly in X , writing (6.1.4)

4(x,,, x,) = 9 ( x n , X)

+ { $(x,,,

x,)

- $( x, x)}

we find that 4 ( x n , x) = &(x,) (say) being convex, is weakly lower semicontinuous. Consequently as n + co, 4 (x, x) < @ 4 (x,, x). Furthermore, , ~ uniformly ~ } bounded. Thus by the since x,+x weakly, { ~ ~ x is hypotheses, 14 (x,, x,) - 4 (x,, x)J+ O uniformly. Hence 4 (x) = 4 (x, x) < lim 4 (x,, x,) = lim 4 (x), and the result is proved. Criteria for Coerciveness Suppose the C ’ functional 4 ( u ) is defined on a reflexive Banach space X and satisfies either of the two conditions:

(6.1.5)

(i) ( 4 ’ ( u ) , u ) 2 g(llull) for some continuous function g ( r ) such that J m g ( r ) / r dr = co; (ii) 4 ‘ ( u ) = Lu + R ‘ ( u ) is a semilinear operator such that m = inf,,u,,=l ( L u , u ) E u,(L), and for llullx sufficiently large, R(u) +

Ilu1l2 2 17(llullri).

where X is continuously imbedded in a Banach space continuous function satisfying q ( r ) + co as r + 03. Then 4 ( u ) is coercive on X . Proof: (i): We will show that norm 1 in X and s 2 0,

lim 4 ( u ) = co as llull+

x with q ( r 00.

) a

For any w of

dt . t Thus by hypothesis, there is a positive number R > 0 (independent of w ) such that 4(sw) - S(0) = / ‘ ( w , $ ’ ( t w ) ) dt = I ’ ( t w , 4 ’ ( t w ) ) 0

0

~ ( s w> )

S(O)+ ~ ( R w ) + - dt.

!J I 4 ( u ) > (2 g ( t ) / t dt + const. Consequently, $f ( u ) is As \lull +do, ~ coercive. (ii): Let N be the finite-dimensional null space of L - m , and let ( u k } be a sequence such that IIuk(lx-+ co. We have uk = u; + u:, where u ; l N and uk” E N . Now since m B u,(L), there is an absolute constant co with g(uk) =

f (J%’ U k ) + R ( U k )

> ~ 0 l l ~ k ’ l l z+x 71(llUklIR),

co

> 0.

6.1

303


Thus if IIuk’llX+ 00, we have 4 ( u k ) + 00. Otherwise, we must have Ijuk’llx < C a n d l l u k ” l j x - + ~ This . implies that ~ ~ u k 0 and L, compact. Thus $ ( u ) can be written as the (L,u, u ) and the functional 4,(u) sum of the convex functional = i(L,u, u ) + “s;c ( u ) , which is clearly continuous with respect to weak convergence. Thus (5.1.3) applies. (ii) Let { g,.,(x) I i, j = 1, 2, . . . , N ) be smooth functions, a n d consider the functional

+

zys,g,(&tj

over the space of functions W, = GI,*[(a,b), R N ] ,where a positive definite quadratic form in E. To apply (6.1.3), let

is

For fixed y , 4 (x, y ) is defined on W , and convex in X . On the other hand, if yn+y weakly in W,, gji(yn)-+gi,(y) uniformly on [ a , 61. Thus for an

where the convergence is uniform over bounded sets in X . Another useful criterion for the attainment of the infimum of a C ’ functional 9. ( u ) defined on a Hilbert space X , that can be stated independentb of semicontinuity assumptions is obtained by requiring 4 ( u ) to satisfy the following “compactness condition.” Condition ( C ) If a sequence (x,) E X is such that 4! (x,) is uniformly bounded and 4 ‘(x,) -+0, then x, has a convergent subsequence. In fact the following result holds: (6.1.1’) Theorem Suppose that the C ’ functional 4 (x) defined on a

304


Hilbert space X is such that 4 '(x) satisfies a uniform Lipshitz condition and is bounded below. Then, if 4 (x) satisfies Condition (C), inf, !I. (x) is attained at 7 and 4 '(2)= 0. Proof: Suppose c = inf, 4 ( x ) is not attained so that c is not a critical value for 4 (x). Then Condition (C) implies for some c > 0, I"+' = { x I (x) < c E } also contains no critical points. (Indeed otherwise there would be a convergent sequence of critical points { x,} with 4 (x,) tending to c and so Condition (C) would imply limn*m x, = X is a critical point with 4 (X) = c.) We now apply the method of steepest descent of Section 3.2 to this situation by considering the initial value problem

+

-dx=

- $'(x),

dt

where xo is an arbitrary point of 4'". By the results of (3.1.27), the solution x ( t ) of this initial value problem exists for all t , provided x ( t ) stays uniformly bounded, and moreover along x ( t )

so that, since 4 ( x ( t ) ) is bounded from below, (1 4 '(x(t))ll +O as t + co. Next we make use of the fact that 4 (x) satisfies Condition (C) to conclude that for any sequence t, + 00, since 4 ( x ( t , ) ) must be uniformly bounded x ( t n ) has a convergent subsequence, x ( t ) with limit X and by the continu"1 ity of 9. ' ( x ) , 4 '(X) = 0. Consequently X is a critical point of 4 . This is the desired contradiction since X E T+', by virtue of (*).

6.1B An illustration A simple (yet nontrivial) example of Theorem (6.1.1) is obtained by considering the T-periodic solutions of the nonautonomous Hamiltonian system

(6.1.7) i = V U ( x ,t), where x ( t ) is an N-vector and U ( x , t ) is a C ' real-valued function of x and t . Supposing that U ( x , t ) is T-periodic in t , we seek T-periodic solutions of (6.1.7). In fact we can prove (6.1.8) Theorem If the T-periodic function U ( x , t ) has the coercive property that U(x,t ) + w as JxJ+oo uniformly in t , then (6.1.7) has a T-periodic solution that can be obtained as the minimum of the functional (6.1.9)

4(x) =

iT{ + Jx2

U ( x , 2 ) ) dt

over all T-periodic C ' N-vector functions

x(t).

6.1


305

Proof: Let W , denote the space of all absolutely continuous T-periodic N-vector functions such that I i ( t ) 1 2 E L2[0,TI. W , = W , ,2[(0. T ) , R N is ] a Hilbert space with respect to the inner product

x(1)

7

(6.1.10)

(x,~)~,=

+

( . t ( t ) . y ( t ) x ( / ) . y ( t ) } dt.

.O

We shall show that (6.1.1 1)

inf 9 ( x ) WN

=

inf wN

IT(f

i2(r)

+ U ( x , t ) } dr

0

is attained at X ( t ) E W,. Then by the results mentioned in Section 1.5, X(l) will be a C 2 function. Consequently, .?(I) will be the desired T-periodic solution of (6.1.7). To verify that inf 9 ( x ) over W , is finite and is attained at X ( t ) , we apply Theorem (6.l.l), and verify the weak lower semicontinuity and coerciveness of 4 (x). First we observe that if x, + x weakly in W,, x, + x uniformly on [O. TI, by Sobolev's imbedding theorem. Thus 9 (x) is the sum of the convex quadratic functional j $ i 2 ( / ) and the (sequentially) weakly continuous functional j$b'(x, I). (6.1.3) ensures that 'J(x) is weak lower semicontinuous. To prove the coerciveness of 9 ( x ) , we let

( L x , x)

(6.1.12)

=i7i2

R ( x ) = I T U ( x , 1 ) dt.

and

Clearly the operator L so defined is a self-adjoint Fredholm operator and rn = inf(Lx, x) over llxll = 1 is zero. Furthermore, Ker L consists of the constant N-vectors ( c ) . so that a general ) mean value element x ( t ) of W , can be written uniquely as x ( t ) =,y(t) c, where ~ ( r has zero on (0, T ) . Now

+

( 6 .I . 13)

IlY(t)ll w,

f k"l}.

a

Then Inn c: < IIy,(t)ll~,< const. C, (say), so that ~ ( 9 ,G ) 4Co/c,'. The complement 9,' of S2, has measure p(S2,') > T - 4C0/c,2. Let n be so large that ~ ( 9 " > ' ) T / 2 . Then on 9'.

1%

= IYn + c,/

lc,I - lYnl 2 IC"l/2.

By hypothesis there is a function v ( r ) such that q ( r ) + w as r + co,with U ( x , 1 ) > ~ ( l x l ) and ,

Thus R(x,)+ proved.

w as c, + w . We conclude that 9 (x) is coercive on W , and the theorem is

306


6.1 C

Minimization problems associated with quasilinear elliptic equations

The abstract considerations of the previous subsection do not distinguish between the well-known significant differences in concrete variational problems involving single integrals on the one hand and multiple integrals on the other. Indeed, for a large class of “regular” variational problems of the form 4 ( u ) = j b , F ( x , u, u,) dx (with u ( x ) an N-vector function of x ) , one can establish both the existence and minimizing property of any critical point i ; ( x ) of 4 ( u ) in the small. Consequently the search for critical points of 4 ( u ) in the large can be decomposed into a succession of local problems. For variational problems involving multiple integrals, this cannot be done. Thus the length of a curve may be defined by polygonal approximation, while the area of surfaces may not be approximated by the analogous simplices, as was mentioned in Section 1.1. Furthermore, the regularity properties of an absolute minimum of functional~involving single integrals are relatively easy to establish, while the analogous regularity properties of minima of multiple integrals remain only partially proven, to date. Critical points of functionals associated with quasilinear elliptic operators, however, do possess a certain interesting “local” minimizing property. Indeed, suppose that 8 is a bounded domain and G ( x ) is a smooth critical point of (6.1.17)

$(u) =

P(x,

Dau, D p u ) dx,

where 1aJ< m - 1 and 1 PI = m over the class of functions in C m ( Q that ) vanish on a8 together with all their derivatives of order < m - 1; while the function F ( x , y , z ) is of class C” and is strictly convex in z for fixed x , y (so that the Euler-Lagrange equations associated with 4 are elliptic). Then i; has the following minimizing property: (6.1.18) Suppose ~ ( x () # 0) is a C“ function vanishing outside of a sufficiently small neighborhood Qx0 of an arbitrary point xo E 8. Then 4 ( 6 + q) > 4 (ii). Consequently, k can never be a relative maximum. Proof: Since 4 ’ ( u )= 0, Taylor’s theorem shows that for some t E [0, 1) (6.1.19)

4(ii+v)= 4 ( i i ) + $ ( 4 ” ( u + t q ) q , q ) ,

where

+

We show that 9 (ii 17) > PI (i;) by proving that the second term in (6.1.19) is strictly positive. To this end, we note for ( a (< m , ll~ll,, < E ( Q ~ ~ ) ~ ~ ~ ~ \

6.1

307


where c ( Q , J + O as p(S2,J-fO. Thus a simple computation shows that for p(D,J sufficiently small, there are constants c,, c2 > 0, independent of n such that

(g

"(u

+ Q)V? 17)

2

c I I I ~ I I Z-, ~ 2 I I ~ l21 t r n - t

Next we take up the problem of finding a function attaining the infimum of the functional Y ( u ) = l n F ( x , D"u, D"u) dx, la1 < m, defined over a bounded domain D c RN with smooth boundary as2 that satisfies the Dirichlet boundary condition Dnuld12= J ; l ( x ) for la1 < m - 1. A wellknown result in this direction is (6.1.20) Theorem Suppose that there is a functionf(x) in W,,,,(D) such that the trace of D*f on d D coincides with f a , la1 < m - 1. Furthermore, suppose that the function F ( x , y , z ) and its partial derivatives d F / d y , d F / d z are continuous, and F satisfies the two conditions: (6.1.21a)

F ( x , y , z ) 2 coIzIP - c I , where co, c I are constants

(6.1.21b)

F ( x , y , z ) is convex in z for fixed x , y .

> 0;

Then infs(u) over the class CS = { u I u E Wm,,(Q) such that D"u I an = f a , la1 < m - I } is finite and attained by a function i ( x ) E 2. Proof: First we note that the class (? is nonempty since by hypothesis f ( x ) E CJ. Secondly, since $ ( u ) and 9 ( u ) + c I have the same critical points, we may suppose that (6.1.21a) holds with c I = 0. Also, hypothesis since !I ( u ) > c,,JnlDmuIP, (6.1.21a) implies that 't ( u ) is coercive on while by Sobolev's theorem, there is a positive constant c, independent of u for la1 < m - 1 such that

e

IIDaull$

< IlD"(u -f)llLp + IIDafIIr, < cJDn'u/ILp+ const.

C,IIDrn(U

-f)ll$+ IlD"fll$

~ , { ~ ~ I P , S m l l D p u I l ~ ) '00, / P - and f in Hence S ( u ) + cc whenever I 1 ~ l l = addition, 4 ( u ) is bounded from below. Thus the argument of (6.1.1) is applicable once we show that by hypothesis (6.1.21b), 4 ( u ) is lower semicontinuous with respect to weak convergence in Wm,,(Q).Let u, + u weakly in Wm,4(Q),with u,, E Then u E e , and for la1 < m , Dau, + D strongly in L,(Q) so that { u,,} has a weakly convergent subsequence (which we relabel {u,,))converging almost everywhere to u in Q . By Egorov's theorem, given c > 0, there is a set QC c D such that D "u, + D "U for (a1 < m - 1 uniformly on Q C , while p(QC) 2 p(Q) - c. Let

e.

l"l

m

308


Since u E Wm,,(S2), p ( 9 - 3,,,)+0 as E + O and N + co. With these preliminaries, we now apply the convexity hypothesis (6.1.21b) to prove the desired lower semicontinuity. Defining

we have

4€,N

( 4

-

4€,N ( U )

=

( %, N ( % , u,)

-

4) N(U7 u ) 1

g,, N ( % ,

+ { g,, d u n , u ) - 9,.

*

By hypothesis (6.1.2 1b), gc,N(U,,

u,) -

Sr,N(Un,

u)

>J F , ( x , DUUn,DrnU){DrnUn-

DrnU),

N

while on

Q2,,"

F ( x , D%,, Dmu)+ F ( x , Dau, D m u ) uniformly

and Fz(x, D%,, D m u ) + F z ( x , Dnu, Dmu) uniformly. Consequently, since Dmu,+ D"u weakly in Lp(S2),lim 4,, (u,) = 4€, ( u ) as n+ 00. Now since F ( x , y , z) is nonnegative, S,,(u,) < 4 (u,); and because c and N are arbitrary, &r~ 4 (u,) 2 4 ( u ) as n + 00. Thus (6.1.20) is established. As already mentioned, general results ensuring the regularity of the minimum i i ( x ) obtained in (6.1.20) have not been found if both rn, N > 1. Thus, in general, one cannot say that i i ( x ) satisfies the resulting EulerLagrange system

2 la1

( - i ) ' " ' ~ * ~ , (D xU ,, . . . ,

mu> = 0,

~ , u =fa, l ~ ~

G rn - 1,

m

where F, = aF(x,xp)/axa.

Recently, the case m = 1 has been successfully resolved for all N , and has been the subject of several books (for example, Morrey (1966) and Ladyhenskaya and Uraltseva (1968)). For the case N = 1, the problem is considerably simplified. Indeed, i i ( x ) is absolutely continuous and in 6 ) for p > 1, and i i ( x ) is Holder continuous with exponent Wm,p(a, rn - l/p. This result was obtained in Section 1.5. To proceed further two simplifying assumptions are useful: first that the Euler-Lagrange equation 4 ' ( u ) = 0 associated with the functional 4 ( u ) is semilinear, and secondly that the equation 4 ' ( u ) = 0 is of second order. The significance of the semilinearity is twofold in that (i) the smoothness

6.1

MlNlMlZATlON PROBLEMS

309

of any generalized solution can often be reduced to the regularity theory for linear elliptic equation (as described in Section 1.5) and moreover (ii) the lower semicontinuity and coerciveness criteria of (6.1.3) and (6.1.5) are readily applicable. Thus, for example, if

4(u) = 4 (Lu, u) + %(u);

u E @m,2(3),

where (Lu, u ) =

1

aap(x)DauD p u

(2 l a l . I b l < m

is a quadratic form associated with the linear elliptic operator L and %(u) = JnF(x, u ) with F ( x , u ) a C’ nonnegative function bounded below by a fixed parabola P ( u ) = c I u 2+ c2 with c I > 0. Then assuming Q c RN a bounded domain, the coefficient aap(x) smooth in 3, the weak lower semicontinuity of 4 ( u ) on Wm,,(3) will follow from Gbrding’s inequality (1.4.22) applied to (Lu, u) and Fatou’s theorem applied to %(u). To see this latter fact, observe that if u, + u weakly in km, ,(a), u, -+ u in measure on 3, and the nonnegativity of F ( x , u ) implies lim? ,m J O F ( x ,u,) > J n F ( x , u ) by Fatou’s theorem. The coerciveness of 9. ( u ) is more delicate, but will follow from (6.1.5(ii)), X = L2(3),provided the positive constant c1 associated with P ( u ) dominates ‘the negative spectrum of L. A key point here is that no growth restrictions need be imposed on the function F ( x , u). The reason for this fact is the nonnegative assumption on F ( x , u ) which ensures the integrability of F ( x , u ) for u E Wm,2(3)and the fact that is some finite number c, say. We need only if inf 4 ( u ) over Wm,2(3) n g - ’ ( c , c + E ) for any fixed consider 9. ( u ) defined on the set Wm,2(3) E > 0. Variational problems associated .with second-order quasilinear elliptic equations possess special simplifying features. These are primarily due to special properties of the functions in W l , p ( 3 and ) to well-known methods (such as the maximum principle) for obtaining a priori bounds for the solutions of such systems. We end this subsection by pointing out a simple (yet useful) instance of this simplification. (6.1.22) On A Priori Bounds Suppose that G(x) minimizes the functional 9. ( u ) = j O F ( x , u, Du) dx over the class C? of functions u in W,,,(O) ( p > 1) that have the prescribed boundary condition u J a n=f. If there 1s a number k > 0 such that for all x E D and I z ( > 0:

(i) F ( x , y , z ) > F( x, k , 0) whenever y > k ; (ii) F ( x , y , z ) > F(.r, - k , 0) whenevery < - k ; (iii) ess sup I f 1 < k over dQ2; then ess sup lii(x)l

0 the equation satisfies the hypotheses of (6.1.22) yith K = supo(l/J g(x)l) so that if 4 , ( u ) = lo(c21 V uI2 - u2 + f g2u4), then inf 5 ( u ) over W , , , ( 0 ) is attained at uI(x, z) E C2(Q) (say) and uI(x,c) satisfies (6.1.30). Furthermore, for c2 > l/A,, by the variational characterization of A,, 4 0 for smooth u ( x ) 3 0 in @'l,2(0). Thus for z 2 > l/A,, q(n, c) 3 0. On the other hand, for t2< l/A,, 9, (uI) < 0 for u , the positive eigenfunction associated with A,, while for u E Wl,2(0), IuI E WI,,(P) and 4 , ( u ) = 4 , (lul). Consequently, (we may suppose) ul(x, c) > 0 in D for c < l/Al. Thus it remains to prove that u,(x, c) is the unique positive solution of (6.1.30) and that ul(x, c) depends continuously on c. (6.1.32) Lemma The solution u I ( x ,e) of (6.1.30) is the only positive solution and depends continuously on c for c E (0,A; ' I 2 ) . Prool: Assume that uI and u2 are distinct positive solutions of (6.1.30) for fixed c 2 E (0, h; I). Then, with h = c - ~ , the difference u = u , - u2 satisfies the system (6.1.33)

Au - Ag2(u:

+ ulu2 + u:]u + pu = 0

in

0,

vlas = 0,

for p = A. Regarding (6.1.33) as an eigenvalue problem for p, with A fixed, denote the smallest eigenvalue of (6.1.33) by p , . Similarly, regarding the positive solution uI as an eigenfunction of the system

for v = h, denote by Y , the smallest eigenvalue of (6.1.34) in which h is fixed. The variational characterization of the lowest eigenvalues p, and Y , implies that p , > Y,. On the other hand, positive eigenfunctions of (6.1.34) belong to the lowest eigenvalue, we have A = Y,; and, by the definition of p , as a lowest eigenvalue, p , G A. Therefore, p1 G Y,, and we have a contradiction and the conclusion is that D = 0, i.e., that u , = u2. Finally, we prove in two steps that u I ( x ,c) depends continuously on c. First, suppose cn+ E , E (0, Then u I ( x ,c,) satisfies (6.1.30), and consequently the two sequences (IAu,(x, e n ) \ ) and ( l u l ( x , 0,

6.2 SPECIFIC MINIMIZATION PROBLEMS FROM GEOMETRY A N D PHYSICS

313

6.2 Specific Minimization Problems from Geometry and Physics

In order to sharpen the abstract results of the preceding section, we now take up some specific minimization problems of importance in differential geometry and mathematical physics. 6.2A

Hermitian metrics of constant negative Hermitian scalar curvature

The classic uniformization theorem for Riemann surfaces discussed in Section 1.2 implies that any C " compact Riemannian 2-manifold ( DX, g) admits a conformally equivalent metric S of constant negative Gaussian curvature if and only if the Euler characteristic of M , x(%) is negative. We proved this result in Chapter 5 as an application of Hadamord's theorem (5.1.5). Here we shall prove an analogous result (for which (5.1.5) is not applicable) for compact complex Kahler manifolds of higher dimension, by replacing the Gaussian curvature with an appropriate scalar curvature function. For such a complex manifold "X , the necessary and sufficient condition we find depends only on the sign of the integral of the appropriate scalar curvature function for ("X , g). Consequently, the result is a direct generalization of the case for complex dimension 1, by the classical Gauss-Bonnet theorem since our result is generally independent of g. The proof we give rests heavily on the minimization methods and the global theory of semilinear elliptic partial differential equations on manifolds. The main difficulty in the higher dimensional cases consists in finding substitutes for the appropriate Sobolev imbedding theorems which fail in this context, for dim, 9R > 1 since then exp u is not integrable for arbitrary u E W , ,2(L;7R,g). (I) Formulation of the problem as partial differential equations Let 9 Z be a C" complex compact manifold of complex dimension N with a Kahler metric g defined (in local coordinates) by setting ds2 =x.9,pg,,p dz" dyp. Then if u is a real C" function defined on M , we consider the Hermitian metric j defined by setting ds2=eZ0ds2.Then we shall find that the Hermitian scalar curvatures R and I? of (m,g ) and g), respectively, are related by the formula

(a, (6.2.1)

k = e C 2 " { R- N

Au},

where A denotes the associated real Laplace-Beltrami operator defined on

3 14

6


( M , g). This formula is derived as follows. Rcjative to the Hermitian connection,' the components of the Ricci tensor Rap relative to ( M , g) are given by the expression

Since 6 = e2"'G, we find that in terms of the components of the Ricci g) tensor R,, for

(a,

Since the desired scalar curvatures are the traces of their respective Ricci tensors, we find (as in Appendix B)

(6.2.3) k e 2 a = R - 2NOa, whereO= 4 A is the "complex Laplacian" relative to (%,g). Clearly (6.2.3) yields (6.2.1). Thus if c2 is some positive (nonzero) constant, the determination of the conformal metric with (Hermitian) scalar curvature - c2 can be effected by proving that the partial differential equation N Aa - R - c2e20= 0 (6.2.4) has a globally defined C solution a over (%, g). Clearly, if a ( x ) satisfies (6.2.4), we find

by integrating (6.2.4) over 9R.Thus an immediate necessary condition for the solvability of (6.2.4) is that j E m R ( x )dV < 0. In fact, this condition is also sufficient and we state

(6.2.5) Theorem A necessary and sufficient condition for a compact Kahler manifold (Em, g) to admit a conformally equivalent (Hermitian) metric with scalar curvature a negative constant is that J - R ( x ) dV < 0. Proof: Since the necessity of the condition was proved above, it suffices to prove that (6.2.4) is solvable for some c # 0 provided the mean value of R ( x ) over 312. is negative. To this end, we proceed to prove the following three lemmas. Lemma (a) The infimum of the functional

'

This is the unique connection that is compatible with the complex structure of Gm and differs from the Levi-Civita connection associated with the Riemannian structure of (%, g) if dimc% > 1.


over the class of functions Wl,2( ?X,g ) = { u is attained by an element U E Wl. 2( %, 8).

315

I lLm (I V uI2 + u 2 ) dVg < co}

Lemma ( ,8) The function U described in Lemma ( a ) can be chosen to be essentially bounded provided sup,,, R ( x ) < 0. Lemma (y) The function ti described in Lemma ( a ) can be chosen to be a C" function and U satisfies the equation (6.2.4). We now proceed to prove these three results. Clearly on combining these lemmas we shall obtain a proof of the theorem whether or not sup,,, R ( x ) < 0.

Proof of Lemma ( a ) To prove inf 4 ( u ) over Wl, 2( % , g) is attained, we use (6.1.1). To achieve this we use the Hilbert space structure of WIT2( Lm , g ) in the following steps; (i): We first show 4 ( u ) defined by (6.2.6) is bounded below on H = W l , 2 ( 9 R ,g ) by 17 (say), q > - co. Then, if 6 = inf, 4 ( u ) , set 4 s + l = { u I u E H , 4 ( u ) < 6 + 1 ] and apply (6.1.1) to 4,+ by verifying that (ii) 4 ( u ) is coercive on gs+ ,, and (iii) weakly lower semicontinuous on g,+ I , to is sequentially weakly closed. show that To show that 4 ( u ) is bounded below on W l , 2 ( c X g) , we note that on setting u = uo + U, where uo has mean value zero over ( " X ,g) and ij = (VOl(!R, g))-lJ"3Ku d V ,

,

4(u)=

s 11

N V uOl2+ J ' R ( x ) u o + U J R ( x ) +

4 c2e2'Je2uo dV.

Thus by combining Poincare's inequality 11 V ~ ~ 1 1 2~ cII. uoI/o, and the fact that e2'0 > 1 2u0, we find from the Cauchy-Schwarz inequality that for any E > 0,

+

(6.2.7)

4(u) 2

N 1 I1 V ;llR(x)ll;,2 - V .I;,* + E ~ ( x d~ ) + $ c2e2; VOI(%, g). C

.II:,2

-

CEll

.ln

Thus setting

E =

N/2c. we find CL

' { ( u ) > - l l R ( x ) l l ~ ~ 2q (+U ) with q ( U ) + c o as ( t i I + c o , (*) N since J R ( x ) < 0. Consequently, inf, 4 ( u ) > - 00. (ii): In the same way we can verify that 4 ( u ) is coercive on 4s+l, where S = inf, 4 ( u ) . To this end, let u, E Wl,2( 9 , g) be such that then we show that S(u,)-+co. Thus we note that an equivalent norm on W , , , ( % , g) can be chosen to be \ \ u I I ; . ~ = 1) V uol12 + !GI2. Hence we need only investigate the behavior of 4 ( u n ) as either 11 V u(Io,

316


+ co with lUnI2bounded or (U,J2-+ 00. In the former case the result follows from (6.2.7) by setting CE = t . In the latter case we use (*). (iii): The weak sequential lower semicontinuity of g ( u ) on 4,+, follows easily. Indeed, if u,, + u weakly in W,,,( %, g), u, + u strongly in I,,(%, g ) by Rellich's lemma. Thus ( R ( x ) u , + ( R ( x ) u . Consequently, by Fatou's theorem and (6.1.3), lim 4 (u,) > g ( u ) . From this fact one also observes that in fact 4,+ I is weakly sequentially closed. Proof of Lemma ( p ) We shall prove the existence of a finite real number k > 0 such that for any element u E 4,+, the truncated function

(6.2.8)

u, =

{

> 0, u < 0,

inf(u, k )

for u

sup(u, - k )

for

,

,(a,

is such that 9 (&) < ( u ) . For then uk E g,+ n W,, g ) , and so the minimizing sequence { u,} must also. Hence ii of Lemma (a)can be chosen to consist of functions with ess sup,,,, J u J< k . To find such a real number k we first observe that if u E Wl, ,(%, g ) , uk E WI,,(G3R,g), and in fact (1 V ukllo, < (1 V uIl0,2. Therefore, it suffices to consider the effect of altering the function u to uk for the functional

,

Since 317, is compact, and sup- R ( x ) < 0, there are two positive numbers a , and a, such that - a , < R ( x ) < - a,. Hence as u + + 00, the integrand f(u) = R ( x ) u 3 c2eZu+ co. Therefore there is a positive number k , such thatf(k,) < f(u) for u > k , . On the other hand, if u-+ - co,f(u) also tends to 00 since sup, R ( x ) < 0. Thus there is a positive number k, such that f ( k 2 ) < f(u) for u 6 - k 2 . Consequently, the desired positive number k in (6.2.8) can be chosen to be sup(k,, k,).

+

Proof of Lemma ( y ) We divide the proof into two cases: Case Z: sup- R ( x ) < 0. We use Lemma (/I) to assert that U is essentially bounded. Hence if u is any C" function defined on ( G x , g), the minimality of Z implies that

(6.2.9)

1 { 4(ii lim c+o

+ED)

- I(ii))

E

-J%{ N v ii, v ti + R ( X ) D + c2e2uu) = 0. Since the C m functions are dense in Wl,,(a, g), we find that the integral identity on the right-hand side of (6.2.9) holds for all D E W,,,(%,g). Thus ii can be considered as a weak solution of the linear nonhomogeneous equation in w (6.2.10)

N Aw = R

+ c2e2'.


3 17

Since the right-hand side of (6.2.10) is in L,,(%, g ) for finitep > 1, the Lp regularity theory for linear elliptic equations implies that U E W 2 , pGx ( ,g ) for all 1 < p < 00. Consequently by the Sobolev imbedding theorem (after a possible redefinition on a set of measure zero) U E Cl,a(Em,g ) (the space of functions with Holder continuous first derivatives of exponent a). Thus, the Schauder regularity theory applies to (6.2.10) since now w = U satisfies (6.2.10) (in the weak sense) with the right-hand side in Cl,a( 317,, g ) . Consequently, U E C 2 3 aGx, ( g). Thus U satisfies (6.2.4) in the classical sense and iteration of the Schauder regularity theory yields the fact that U E C "( Em, g). Case ZI: sup, R ( x ) > 0. We reduce this case to Case I by the following device. Write a tentative solution u of (6.2.4) in the form u = 0 + w , where (6.2.1 1)

-

N Av - R - c2e2"e2"= 0 with

R = {vol( %, g ) }

-'I

R dV,,

9ll

(6.2.12)

N AW - R ( x )

+ R = 0.

Clearly, (6.2.12) is uniquely solvable up to an additive constant, and since < 0 by hypothesis, the arguments used in Lemmas (a)and ( p ) apply to (6.2.1 1) with w fixed. Furthermore, once a minimizing essentially bounded ij is found for the functional Z(v) related (6.2.1 l), the regularity argument of Case I holds. Since the solution w of (6.2.12) is also a C" function, u = 5 + w is a C function, and clearly satisfies (6.2.4), as can easily be seen by adding (6.2.1 1) and (6.2.12). Now we interpret the condition l a R ( x ) dV < 0 in terms of analytic invariants of (%, g). In the case of one complex dimension, the Gauss-Bonnet theorem implies the Euler-Poincare characteristic of Gx,x(%), is negative. More generally, an interesting formula from the theory of Kahler manifolds implies that

l x R ( x ) dV

=

k N / cI r \ w N - I ,

k, = positive const.,

"X

where c I is the first Chern class of have

'-3Rand

w is

the fundamental form of

(Gx,g). Thus we

(6.2.13) Corollary A necessary and sufficient condition for a compact Kahler manifold

(Gx,g) of complex dimension N to admit a metric 2, conformal to g, with constant negative Hermitian scalar curvature is j:mcl A a N -' < 0. For algebraic varieties of L91t this condition is expressible in terms of the sign of the degree of the canonical divisor of

a.

6.2B. Stable equilibrium states in nonlinear elasticity

Generally speaking the equilibrium states of an elastic body B acted on by given conservative forces can be determined as the critical points of an appropriate smooth potential energy functional 4 (u). Because of the great complexity of the possible equilibrium states, it is important to determine

318


those states that correspond to absolute minima of 4 ( u ) . Indeed, by the remarks of Section 4.3B, such states will be stable. Here we take up the problem of demonstrating the actual attainment of the infimum of 4 ( u ) for various problems in elasticity. Case I : Deformable plates: We suppose a thin flexible elastic plate is clamped along its edge and is acted on by the combined action of forces acting along its edge and a force f acting normal to the plane of the plate. The resulting equilibrium states are governed by the von Karman equations (1.1.12). Our discussion of Chapter 2 shows that these equilibrium states are solutions of the operator equation W,(U) = f, where f E W2.,(a) is a representation off and is proportional to the magnitude off. Moreover in (2.5.7) and (2.7.18) we showed that the operator

&(,

U) =

u

+ CU - ALu

is (i) a gradient mapping of W2,2 ( Q ) into itself with

W)= 4 llUll;,2

- (h/2)(Lu9 U) +

ii (Cut ).

so that ',(u) = W,(U), and (ii)d,(u) is a proper-mapping (in the sense of Section 2.7). Analogous results clearly hold for g,(u) 2g,(u) - ( f , u). Thus to demonstrate the coerciveness of 4,(u) on W2,2(L?) let ((u,(j+ co. Then q,(U,)

=

11~,112 + 4 (Cu,,

U,) - A(J%,

Un),

- (f, U,)

t

= I I U ~ I+I ~ I I C ( ~Un)I12 ~ , - ~ ( ~ ( u n , ~ 0 -) (f, ~

Consequently for any c

> 0, by

the Cauchy-Schwarz inequality

+ ( t - Xc)IIC(u,, Choosing f = Xc we find S",Cu,) 2

-

- ( ~ / 4 1 1 ~ o l 1 2- 2llfll

llUnl12

&(%)

II~,1I2

n ) .

- 2llfll

IIU,II

-

ll%ll.

2~211~0112.

Thus 4,(un)+ 00 for fixed X as n + 00. The weak lower semicontinuity of g,(u) is an immediate consequence of the criterion (6.1.3) and the complete continuity of the operators L and C. Thus from (6.1.1) we conclude (6.2.14) For any fixed h and f, the functional !r,(u) is bounded below on W2,,(Q) and attains its infimum on that set. Moreover this infimum yields a smooth solution of the associated von Karman equations. Case I I : Deformable shallow shells: We now generalize the result for plates just obtained by supposing that instead of being flat, the thin elastic structure S under consideration has some initial curvature described by the functions k , ( x , y), k2(x,y ) that measure the Gaussian curvature of 5'. We suppose that the shell is acted upon by forces on its boundary as well as by forces Z acting normal to the shell. The resulting deformations can be determined by the nonlinear von Karman equations (4.3.1) 44.3.2) together with the boundary conditions


3 I9

(4.3.23)-(4.3.24). Again, for simplicity, we suppose that Z = A$,, and that AF, is a solution of the linear problem A2F = 0 together with the boundary conditions (4.3.23)-(4.3.24). Then solutions (w.f ) can be sought in the form ( w ,F + AF,), where w and F satisfy (6.2.15)

A ~ F =-

2 I

1

3

WI -

(k,w,), - (k2WyIy,

+ h[F,, w] + (kiF), + (kiF), + AZ',

(6.2.16)

A% = [ F , w]

where Z' = Z

+ (k,F,), + (k,F,),

(6.2.17)

D"FI,, = D'%l,, = 0

together with the homogeneous boundary conditions for

la1 G 1.

As in (2.5.7), these equations can be written in the form of operator equations in (9

F=

-

(ii)

w

C ( F , w)

=

k,,*(a):

'2 C ( w, w) - Lw,

+ AC(F,,

w)

+ Lw + AZ'.

Substituting (i) into (ii), and using the results of (2.5.7), we find that the associated potential energy functional can be chosen to be (6.2.18)

e ( w , A) = 11w'112+

I[ ;C ( w , w)

+ L i W l [ 2 - WLW, w ) - A(Z', w).

We now will show

+

(6.2.19) Theorem For all Z , and functions $, k , , k,, inf 9(w, A) over k2,2(Q) is finite and attained by an element of W,.,(Q) that can be associated with a solution (w, F ) of (6.2.15)-(6.2.17). Proot: We apply (6.1.1) and show that 9 ( w , A) defined by (6.2.18) is weakly lower semicontinuous and coercive on k,,,(a). The weak lower semicontinuity on k,,,(a) is obvious by (6.1,3) since by (2.5.7) the operators C(w, w), L,(w), and L are completely continuous on W,, ,(a), which means that 9(w. A) is the sum of a convex and a weakly sequentially continuous functional. Again it remains to prove the coerciveness of 9(w, A). To this end, consider the set

(6.2.20)

El = ( w I llwll

=

I , I ( w ~-~ A(Lw, ~ w) < $ }

Clearly on the weak closure of El, E ; , infE,,l/w(l> 0, for otherwise there would be a sequence w,, E El such that w, + 0 weakly, so that by the complete continuity of L, (Lw,, w,,) +0 and hence IlwJ < 1 for sufficiently large n . Thus on E ; , inf /I C(W, W)l12 > a2 > 0 (since otherwise there would be a W E El. such that C ( W , W ) = 0, so that the surface W = W ( x , y) has zero Gaussian curvature a t . and on a a, w = 0, which implies W = 0).Now setting w = IIwIIu with 11011 = 1, we have for w E El, as llwll --* w .

where K is a constant independent of w and u; and for u p El , (6.2.22)

G ~ ( w A) , > Ilwll2(ll~ll2- ~ ( L uu)) ,

-

A(Z', W)

> 4 l l ~ 1 -1 ~K .

Hence in both cases, G,(w, A)+w as IIwII+w.

6.2C Plateau's problem Here we solve the (simply connected) Plateau's problem for recitifiable Jordan curves r c R3, (cf. Section 1.1A) as a modification of the ideas

320


discussed in Section 6.1. More precisely, we seek a smooth, simply connected parametric surface S in Iw3 spanning r such that the area of S is minimal. Thus if we let G? be the open unit disk in Rz, we seek a vector r ( x , y ) = ( u , ( x ,y ) , u2(x,y), u,(x, y ) ) that represents a surface S spanning r in such a way that (1) a W is continuously mapped onto r in a one-to-one manner, and (2) the area of S , (6.2.23)

A ( S )=~

J ( Ql J ( ~ l ,

u2)l2 + IJ(u,, u3)I2+ ( J ( u , ,u3)lz)1’2dxdy

is minimized, where IJ(u, o)l is the Jacobian determinant of u and o with respect to x , y . An important simplification of this problem results from differential geometric considerations. For any surface S = { r 1 r = [ u I ( x , y )u, , ( x , y ) , u 3 ( x , y ) ] } we , write the first fundamental form as ds2 = d r . d r = g , , d x 2 + 2g1, dx dy

+ g2, dy2,

where g,, = r x *rx,

g,, = r; ry, and

g,, = ry*ry,

and the area of S as A ( S ) =JJ(g,1g,,Q

9:2)1’2dx d r *

Since for any three numbers a , p, y (all positive) d a y - pz

0.

+

Step 2: Now we prove

(6.3.6) Lemma Suppose that %(x) satisfies the hypotheses of (6.3.4), then the constraint set C,=(x( ~ ( x ) = R , ( ~ ' ( x ) , w ) = O , w E K e r L ) is nonvacuous for each R

> 0.

Proof: Let S, = { x I (?R'(x), w,) = 0} (i = 1, . . . , N ) . Then the constraints can be written

s,

where ( w l , . . . , w N ) is an orthonormal basis for Ker L and y E [Ker L ] I . Regarding the left-hand side of the above equation as a function of p = (PI, . . , , P N ) and fixingy E [Ker L ] I , (9'(x), wi) = 0 (i = I , . . . , N ) is satisfied by those N-vectors p that are critical points of the functional F( p ) = % ( y + C;"= p,wj>.Thus by the strict convexity of the functional %, there is one and only one critical point p ( y ) = (p,(y), . . . , p N ( y ) )of F( p ) = % ( y + p,wJ for fixed y E [Ker L ] I . Thus for each positive s, there is an element )-(s) = sy + Z;"= pj(sy)wi such that (%'(y(s)),w) = 0 for all w E Ker L. Now for fixed nonzero y E [Ker L ] I the function G ( s , PI, . . . , P N ) = %(sy + I P,W,) is a strictly convex function depi/+ 00, G(x, PI, . . . , p N ) + co. fined on R N + I , such that if Is1 + Hence, as a function of s, g(s) = %(sy + I pj(sy)wl)-+ 00 as Is1 + 00. Therefore the lemma is proved provided we show that the function g(s) is a continuous function of s. This is immediate from convexity theory since g(s) = inf G(s, PI, . . . , p,) over PI, . . . , PN and G is strictly convex.

xy=

xr=

x;"=ll

xy=

Step 3: We show that inf ( L x , x) over C, is attained by an element u, E C,. Since C, is a weakly closed set in the Hilbert space H , by (6.1.1) it suffices to prove that the functional ( L x , x) is weakly lower semicontinuous on H . The weak lower semicontinuity of ( L x , x ) is plainly an

328


immediate consequence of (6.1.3) since L can be written as the sum of a self-adjoint positive operator L , and a compact self-adjoint operator L,. The variational problem conjugate to the one mentioned in (6.3.2) is also useful in the study of nonlinear eigenvalue problems. Indeed for each fixed number R , we consider the level set aA, = { A (x) = R }, and the number C, = supaA, 3 (x), and prove (6.3.7) Theorem Suppose the C‘ functionals &(x) and 9(x) defined on the reflexive Banach space X satisfy the properties: (i) @ (x) is coercive, weakly lower semicontinuous on X,and for each fixed nonzero x E X , the real function f(t) = @ ( t x ) is a nonzero, increasing function of t ; (ii) 3 (x) is continuous with respect to weak convergence and such that 9’(x) = 0 implies x = 0, and for each nonzero x E X , g ( t ) = 3 ( t x ) is a strictly increasing function of t ; then the number C, defined above is a critical point of 3’3 (x) restricted to aA, for each R # 0 in the range of 93 (x), and moreover if 3 (x,) = C, for x, E aA, ( x R , A,) is a nontrivial solution of the. equation A ’ ( x ) = AB’(x). Proof: Since &(x) is coercive, the level set aA, is bounded and so the functional 3 (x), being continuous with respect to weak convergence, is bounded on aA,. Consequently C, = sup 3 (x) over a.4, is finite. Thus any maximizing sequence { x n } E aA, is bounded and so (after possibly passing to a subsequence) x, is weakly convergent with weak limit E. The continuity of 3 (x) with respect to weak convergence implies C, = 3 (X), and we set X = x., Thus to establish the desired result we show x, E &IR and that X, satisfies the equation @‘(x) = A,%‘(x,) for some finite number A,. To this end suppose X, FaaA,, then by the weak lower semicontinuity of &(x) we may suppose & ( x R )< R , and moreover hy) R . Now pothesis (i) of the theorem implies for some f > 1, @ ( f x R = hypothesis (ii) implies that 3 ( t x R ) > cR = supA, 3 (x) (a contradiction). Consequently xR E aA, and by (3.1.31), there are numbers A,, A, such that A , 6 ! ’ ( x R ) + A2’% ’(x,) = 0. Now A, # 0 since otherwise 3 ’ ( x R ) = 0 which implies by hypothesis (ii) that x R = 0, which in turn contradicts the fact that R # 0. The isopeximetric methods for studying nonlinear eigenvalue problems presented above has a distinct limitation, namely they yield only the analogue of the first eigenuectorfor an operator A . In the case of linear operators, one need not proceed to discuss any further critical point theory since the notions of orthogonality and orthogonal complement enable an iterariue isoperimetnc approach to yield a complete set of eigenvectors for A . However for nonlinear operators we cannot proceed to the full analogue of this completeness result until we develop deeper critical point theories. Indeed it will be necessary to find topological “constraints” that enable one to discuss “higher” eigenvectors for nonlinear operators.

Example: In order to understand the content and precision of (6.3.2),

6.3

329

ISOPERIMETRIC PROBLEMS

consider the following semilinear Dirichlet problem defined on a bounded domain' Q c R N : (6.3.8)

Au

+ k ( x ) u + hg(x)u"

= 0,

u l a Q= 0, where k ( x ) and g(x) are smooth functions (say Holder continuous with g(x) > 0 on As an application of (6.3.2) we shall prove

a).

+

(6.3.9) Theorem If u lies in the open interval (1. ( N 2 ) / ( N - 2)), then the system (6.3.8) possesses a one-parameter family (uR, hR)of nontrivial smooth solutions for each positive number R # 0 with ( Q u i + l = R and uR > 0 in 3. Proof: Clearly the appropriate functionals U and 93 used in applying (6.3.2) can be defined by setting

@ ( u )=

s,{Iv

u12

- k ( x ) u 2 } dx,

iT3(u)

=

b

g(x)u0+1 dx.

Then assuming for the moment that u is an odd integer, one easily verifies that (i) the functional d ( u ) is weakly lower semicontinuous on @,,2(3), and (ii) for u E (0, ( N + 2 ) / ( N - 2)), the functional 9( u ) is weakly continuous and strictly convex on @l.2(Q), Thus to verify that @(u) is coercive relative to 3 on Pk,.,(a), we first note that by Jensen's inequality for (Y = inf g(x) over

a,

Hence ( \ u (L,(Q) ( is uniformly bounded, if ($3 ( u ) < const. Furthermore, by utilizing (1.3.28), for example, there are absolute constants c I , c2 with c l > 0 such that

Consequently, @ ( u ) + c/3 as lju(l+ oo with ?B ( u ) < const. Since the range of the functional % ' ( u ) is [0, 0 0 ) and ' ( u ) = 0 if and only if u = 0, R = 0 is the only excluded point in the range of 9(u). Thus for every R in the open interval (0, oo), the system (6.3.8) has a family of nontrivial weak solutions ( u R ,hR)with I Q g ( x ) u " + I dx = R . Furthermore, both functionals & ( u ) and 9( u ) are unchanged by replacing u by J u Jand J u JE @,, 2(Q) if u E kI, @). Hence we may conclude that any minimizing sequence { u,,} E (21~ for inf U ( u ) over y3R will be minimizing if {u,,) is replaced by { Iu, , ~ }. Since a subsequence of { u,,} converges almost everywhere to uR, we may suppose that uR 2 0 (a.e. in 3). Furthermore, the linear regularity

'

Later in a geometric context this problem will be considered again with compact Riemannian manifold 'XNof dimension N .

replaced by a

330


theory, as mentioned in Section 1.5, permits us to suppose that uR > 0 (everywhere in and smooth enough to satisfy (6.3.8)pointwise in 52 and at all sufficiently smooth portions of 352. To show that u, > 0 in 52, we use the maximum principle. (See Protter and Weinberger, 1967.)Indeed, if uR = 0 for some x E a, (6.3.8)implies that uR = 0 in a. Finally, to remove the restriction of oddness on u, we note that if we replace the term g ( x ) u " with g(x)lul"-'u in (6.3.Q and repeat the argument just given, we again find a solution uR 2 0 for the amended system. However, in this case, g(x)uj = g(x)lu,l"-'u,; and the result is proven.

a)

There are two possible important extensions of (6.3.9) which immediately come to mind:

+

(i) possible validity of (6.3.9)for a > ( N 2 ) / ( N - 2); (ii) removal of the restriction that be a bounded domain of R". In both cases, the main difficulty is that the functional 3 ( u ) is no longer weakly continuous (or equivalently, 2'% ' ( u ) is no longer completely continuous). Concerning (i), we note that (6.3.9) is sharp in the following sense, as follows from our discussion in (1.2.7).

(6.3.10) Suppose that in the equation (6.3.9)k ( x ) = 0,g(x) = p 2 > 0, and u > ( N + 2 ) / ( N - 2). Then (6.3.8)possesses no nontrivial smooth positive solutions. Concerning the extension (ii), we shall prove later in (6.7.25) (6.3.11) Theorem Suppose 52 = R", g(x) = g ( # 0) a constant, and k ( x ) = k a negative constant. Then the system (6.3.8) has nontrivial smooth positive solutions in Wl, 2(RN) if and only if u < ( N + 2 ) / ( N - 2). In the general case we consider functionals (with la[< m )

a(.) = p ( x , D"u)

and

% ( u ) =LG(x. Dpu)

with 1 PI < m - 1 defined over W,,,p(f2), with s2 a bounded domain in RN. Then provided the functions F ( x ,y , z ) and G(x, y ) satisfy the appropriate ( u ) are C ' func(Sobolev) growth conditions to ensure that @ ( u ) and 91 tions defined on @',,7,p(i2)Theorem (6.3.7)yields conditions for the existence of nontrivial solutions (u,, A,) for J y J< m

2

2

( - I)'"' D"F,(x, D y u ) = A

la1 < m

( - 1)"' DpGp(x,Db),

1PlGrn-l

D"ul,,

= 0,

JaIQ

m - 1.

The major typotheses to be verified include (a) the weak lower semicontinuity in W,,,p(52)of W ( u ) , (b) the continuity of $8 ( u ) with respect to

6.3 ISOPERIMETRIC PROBLEMS

33 1

weak convergence on @,,,+(Q), and (c) the coerciveness of Q ( u ) . The properties of the functions F ( x , D y u ) , G ( x , D y u ) necessary to check these hypotheses have been discussed earlier. 6.38 Solvability of semilinear gradient operator equations

We now turn to the second area of applicability of isoperimetric variational problems mentioned earlier: exact solvability criteria (analogous to those of 5.4E)for inhomogeneous gradient operator equations of the form (6.3.12)

4‘(u) =f

that do not require the linear growth of S’(u) as in Section 5.4E. Here $ ( x ) is assumed to be a C 1 real-valued functional defined on a Hilbert space H . Moreover, 4 ‘ is assumed to be a semilinear gradient operator mapping H into itself so that !I ‘ = L + 9Z ’, where L is a self-adjoint Fredholm operator and % : ’ is a (nonlinear) completely continuous mapping of H into itself. Since in the most interesting cases the operator L will have either negative spectrum or a nontrivial kernel, a solution uo of (6.3.12) will not correspond to an absolute minimum of the functional I ( u ) = 4 ( u ) - (f, u). Indeed, a solution uo will, generally speaking, correspond to a saddle point of I ( u ) . For quadratic functionals 4 ( u ) , such critical points can be reduced to absolute minima by a judicious use of orthogonality. Here we show that in certain circumstances we can find a “nonlinear extension” of orthogonality that is suitable for studying the solvability of (6.3.12) by analytic means. In later sections, we shall show how saddle points of functionals 4 ( u ) can be studied by topological methods. In order to find suitable necessary and sufficient conditions for the solution of (6.3.12), we introduce the notion of “natural constraint.” Suppose a critical value of a C ’ functional 9 ( u ) is not an absolute minimum over a Hilbert space H , but there is a submanifold 9R of H such that: (i) c = inf, 4 ( u ) is attained by an element U E 9TL ; (ii) for any U E “li, n V l ( c ) , g ’ ( U ) = 0, so that U is not only a critical point of $ restricted to UX; but also of 4 considered as defined on H itself; (iii) every critical point of 4 ( u ) lies on ‘X . Such situations are common in geometric problems whose solutions must satisfy fixed geometric side conditions (see for example Section 6.4B below). The following result gives a fairly general construction for natural constraints associated with a C2 functional (u).

332


(6.3.13) Theorem Let N be a closed linear subspace of a Hilbert space H , and suppose 4 ( u ) is a C2 functional defined on H . Set S = { u 1 u E H, 4 ' ( u ) l N ) ,

and suppose (a) S is closed with respect to weak sequential convergence and nonvacuous, (b) 4 ( u ) is coercive on S and weakly lower semicontinuous there, (c) 4 " ( u ) is definite on N for each u E S . Then, (i) c = inf, 4 ( u ) is finite and attained by an element U E S , and (ii) S is a natural constraint for the functional 4 ( u ) (on H ) . Proof: (i) follows immediately from hypotheses (a), (b), and the result (6.1.1). Consequently, we need only establish (ii). To this end, we observe that the elements of S are the zeros of the operator P 4 ' ( u ) for u E H , where P is the canonical projection of H onto N . Now the operator P 4 ' ( u ) is a mapping of H into N whose derivative P 4 " ( u ) is surjective. Indeed, by hypothesis (c), P 9 " ( u ) maps N onto itself for each u E H (by virtue of the Lax-Milgram theorem (1.3.21)). Therefore, the result (3.1.37) implies that for an extremal U of 4 ( u ) restricted to S , there is a fixed element w E N , such that U is a critical point of defined on H . 4 ( u ) - ( P 4 '(U), w ) Hence ii satisfies

S'(U)

= S"(ii)w.

Taking the inner product of this equation with w ,we find that, by virtue of the definition on S , ( 4 "(U)w, w ) = 0. Consequently, hypothesis (c) implies that w = 0, so that 4 '(U) = 0. Thus to verify the fact that S is a natural constraint for 4 ( u ) on H , we need only show that every critical point u of 4 ( u ) (considered as defined on H ) lies on S . However, this last fact is immediate since 4 '(0)= 0 and so 4 '(0) is necessarily orthogonal to N . To utilize the above general prescription for constructing natural constraints relative to the operator equation (6.3.12), we make use of the fact that 4 ' is semilinear (i.e., 4 ' ( u ) = Lu %'(u)). Accordingly we choose the subspace N carefully to take account of the spectral properties of L. More precisely, we shall determine a characterization of a tentative critical value c for the functional S.(u) based on the spectral properties of L, and then utilize (6.3.13) and hypotheses concerning the mapping % ' ( u ) to ensure that the number c is actually a critical value for 4 (u).

+

Thus if % ' ( u ) = 0, it is easily verified that: (a) the only possible critical value E of the functional J(u) = f (Lu,u ) - (f, u) can be characterized by the formula: (6.3.14)

F=

inf

sup

xeH+ YEH-

sup reKerL

J(x+y+r),

333


where H + is the linear subspace of H on which L is positive definite, and H - is the subspace of H on which L is negative definite; (b) if L is a Fredholm self-adjoint operator, P is attained and finite if and only iff is orthogonal to Ker L. Now (a) and (b) together imply the exact solvability criterion for the operator equation Lu = f defined on H , and the formula (6.3.14) exhibits the fact that the associated critical value of J ( u ) is generally not an absolute minimum. We now establish (a) and (b). We first observe that if f is not orthogonal to Ker L , J ( z ) = -(f, z ) can take any real value (positive or negative). Now the orthogonality of H , , H - , and Ker L implies

+ a) = J ( x ) + J ( y )+ J(z).

J(x +y

Consequently, if f is not orthogonal to Ker L, ? cannot be finite; and conversely, if flKer L, J ( x + y + z ) = J ( x ) + J ( y ) , so that (6.3.15)

?=

inf J ( x ) + supJ(y). H+

H-

Since the operator L is self-adjoint and Fredholm, there is an absolute constant a that for x E H , any y E H _ (6.3.16)

(Lx, x )

> allxlI*,

> 0 such

( L y , y ) < -allyll*.

Thus by (6.3.16), infH+J ( x ) is attained at some x E H , and the strict convexity of J ( x ) implies that this X is unique. A similar statement holds for supH_J ( y ) and a unique point p E H - since supH- J ( y ) = infH_( - J ( y ) ) . Consequently (6.3.15) implies that? is attained at any point of the form ii = x + p + z for any z E Ker L. Let P, and P- be the canonical orthogonal projections of H onto H , and H - respectively. Then J ’ ( 7 1 ) = LT2 =

-f= (Lss - P , f) + ( L p - P _ f)

P , J ’ ( x) + P- J ’ ( 7 ) = 0.

In the general case, we consider first a functional R ( u ) and define a number c analogous. to C by setting (6.3.17)

c = inf

sup

x€H+y€H_

sup

g(x

rEKerL

g ( u ) = ~ ( L Uu ), +

+ y + 2).

and determine precise conditions on the functional R ( u ) for c to be finite and a critical value for :I ( u ) on H . Then we set R ( u ) = % ( u ) - (f,u ) and show that these conditions yield necessary and sufficient conditions for the solvability of the equation (6.3.18)

Lu

+ 9z ’(u ) = f,

which reduce to the usual orthogonality condition of Fredholm in case !R’(u) = 0. To this end, we prove the following two results.

(6.3.19) Lemma Let R’(u) be completely continuous. Suppose the C2 functional 9 ( u ) = ( L u , u ) + R ( u ) satisfies the following conditions:

+

(i) the set S o = { u

I

4 ’ ( u ) I K e r L } # 0;

334


+

(ii) for fixed u E So, the functional 4 ( u w ) is strictly concave in w E Ker L u H - and 4 ( u w ) + - w as ( j wI/ -+ 6 0 ; (iii) f o r u E S = { u ( 4 ' ( u ) l { H _ u K e r L } } , 4 ( u ) + w a s I ( u l l + w .

+

Then the number c defined by (6.3.17) is finite and is a critical value of 4 ( u ) on H . Using the same notation and setting %'(u) - f = R'(u), (6.3.20) Theorem Suppose the operator GJI. '(u) satisfies the following conditions: (i) for u E Gsrt = { u 1 %'(u) - f l K e r L } , L + %"(u) is negative deas ( I w ( l - + + b o , w E H _ U finite on H - u K e r L and $ ( u + w ) + - w Ker L ; (ii) for u E S = { u I ( L u + % ' ( u ) - f ) l H - u K e r L } , 4 ( u ) + w as

IIuII

+

Then the equation (6.3.18) is solvable if and only if the set Gm is nonempty. Moreover, if solvable, a critical value of 4 ( u ) is given by (6.3.17). Proof of (6.3.20) using (6.3.19): Indeed, if (6.3.18) is solvable, the set 9 2 is necessarily nonvacuous since any solution of (6.3.18) is an element of 317,. On the other hand, if 9 2 # 0,Lemma (6.3.19) implies that the number c defined by (6.3.17) is a critical value for I ( u ) defined on H , and so (6.3.12) is necessarily solvable. Proof of (6.3.19): We shall show that the result follows from (6.3.13) by verifying that the set S = ( u 1 4 ' ( u ) I [ H - u Ker L]} and the functional 4 ( u ) satisfy hypotheses (a)-(c) mentioned in the statement of (6.3.13). First we show that S is closed with respect to weak convergence and is nonvacuous. Since 4 '(u) = Lu R '(u), un -+ u weakly in H implies Lu, + Lu weakly in H and R'(u,) -+ R'(u) strongly in H . Thus un E S implies u E S . We show S is nonvacuous by using the fact that So is. Since So is nonvacuous, for some R, J

+

6 = sup 4(X +J + z ) Ker L

< 03,

while the finite dimensionality of Ker L implies that for some 2 E Ker L , 6 = 4 (K + j j Z). Setting f = 2 + J 2, we consider the functional $,(E + y + z) = - $ ( f+ y + z ) fory E H - and z E Ker L. By hypothesis (ii), 4 , is coercive on H - u Ker L and moreover 4 , is certainly weakly lower semicontinuous there. Now (6.1.1) implies that for J 2, P 4 ; ( E +r" Z ) = 0,

+

+

+

+

335


where P is the projection of H onto H - u Ker L . Consequently, fi +J + i ES. To establish the weak lower semicontinuity of PI ( u ) on 5 , we again uniquely decompose an arbitrary element u E S into u = x + y + z (as above) and observe that PI’(u) = 0 implies that Ly = PR’(u). Thus if u, -+ u weakly, Lyn is strongly convergent since PR‘(un)is. Thus for u E X , the quadratic form is continuous with respect to weak convergence. Accordingly, for u E S , the functional 4 ( u ) can be written (6.3.21)

4 ( u ) = + ( L . Yx) , + (Ly,y)

+ R(u),

and is thus weak lower semicontinuous, by (6.1.3), since the latter two terms in (6.3.21) are both continuous with respect to weak convergence. The coerciveness of $ ( u ) on S is an immediate consequence of hypothesis (ii) of (6.3.19). Finally the definiteness of 4 ” ( u ) on N for u E S is an immediate consequence of the strict concavity of 4 ( u w ) also given in hypothesis (ii). Therefore, by (6.3.13), S is a natural constraint for ( u ) on H , and moreover, inf, 4 ( u ) is a critical value of 4 ( u ) on H . It remains to show that the number c defined by (6.3.14) actually equals inf, 4 ( u ) . This follows from the fact that the points of S are precisely the solutions of the equation

+

P4’(x + y

+ z ) = 0,

u=x

+ y + 2,

determined by letting x vary over H , and characterized for each 4(x w). Thus Lemma fixed x E H , by the critical value s u p wE Ke r L U H(6.3.19) is proven.

+

Before deriving some further consequences of (6.3.12), we first show that in certain cases the hypotheses of that result can be considerably weakened by restricting the spectrum of L . (6.3.22) Corollary Suppose u,(L) is nonnegative, then the coerciveness part of hypothesis (i) of (6.3.19) can be removed provided we suppose that for u E % and some c > 0, 9 ” ( u ) - tL is negative definite on Ker L u H -. Proot: Let u E %, and decompose w = y theorem,

(6.3.23)

+z

with y E H - ,

+ w ) = 4(u) + ( 9 ’ ( u ) , w ) +0/(1 I

9(u

- s)(9”(u

I

E Ker L. Then by Taylor’s

+ sw)w, w ) ds.

Since u E %, (S’(u), z ) = 0; and since L is Fredholm, there is a finite absplute constant y such that (Ly,y)-< - rlly112 for ally E H-. By hypothesis, for some c > 0, 4 ” ( u ) = 9 ” ( u ) cL is such that ( I ” ( u ) w ,w ) < 0. Moreover, u J L ) c [0, m) implies dim(H--uKer L ) < m, so that there is a positive constant a such that for fixed u E %, max,lwll,I( 9 ” ( u ) w , w ) = - 2 a . Consequently, by continuity, there is a p > 0 such that IlS”(h) - S”(u)ll G a

for

Ilh -

u I I < p.

336


Thus for s sufficiently small, llswll

 -

0,

Choosing 6 = yr/2, we find 9 ( u + w ) + - 00 as llwll+ M, as required. As a useful application of the above results, we consider the semilinear gradient operator equation

(6.3.24)

Lu = % ’ ( u )

+g

with II%’(u)(( < const. for all u E H, where % ( u ) is a C 2 strictly convex function of u, that is, continuous with respect to weak sequential convergence and the essential spectrum of L is nonnegative. Then the following result, analogous to (5.4.29), holds.

(6.3.25) Theorem A necessary and sufficient condition for the solvability of (6.3.24) is that the set 9Rg = ( u 1 % ’ ( u ) - g l Ker L ) be nonempty. Moreover, the operator L 3’regarded as a mapping from H into itself has an open range.

+

Proof We leave the proof of the first part of this result to the interested reader since it follows routinely from the facts just established. We show that the mapping L + 3‘has open range, by showing that if Xgo is nonvacuous, so is %,. for I(go- g’l)< e for some fixed E > 0. Let go E Range(L + %’), then by the first part of the theorem, the set

EXgo= ( u ( u E H , ( % ’ ( u ) - g o ) l K e r L } contains a point uo E H. Moreover, for any u E [Ker L ] l , the set

EXw= { u I u E H , ( % ’ ( u ) - go - o ) l K e r L } is nonvacuous since it contains ug. Finally, suppose then we shall show that the set

(6.3.26)

EX, = ( u I u E H , ( % ’ ( u )

-

I

E Ker L has sufficiently small norm,

go- r ) l K e r L }

is nonvacuous since it also contains an element solve the operator equation

U(I)

near uo Indeed, u ( z ) E Xz if we can

(6.3.27) P ~ ( % ’ ( u ( z ) ) )= Po(z + go), where Po is the canonical projection of H onto Ker L. If z = 0, uo satisfies (6.3.27). Thus for llzll sufficiently small, an application of the inverse function theorem shows that if the left-hand side of (6.3.27) is regarded as a mapping of a small neighborhood of the origin of Ker L into Ker L, then (6.3.7) is solvable. Indeed, the inverse function theorem is applicable here since L is a Fredholm operator, dim Ker L < 00, so that the strict concavity of % ( u ) implies that Po%”(uo) is an injective linear mapping of Ker L into itself, and so is invertible. Finally, applying the first part of the our result we find, on the basis of the above paragraph, that for all g E H with 11 g - goll sufficiently small, the equation Lu + % ’ ( u ) = g is solvable. Thus the map L 3’ has open range, as required.

+

6.4 ISOPERIMETRIC PROBLEMS I N GEOMETRY A N D PHYSICS

337

6.4 lsoperhnetric Problems in Geometry and Physics

Here we shall indicate how the abstract results of the previous section can be used to solve various concrete problems in mathematical physics and differential geometry. 6.4A

Families of large amplitude periodic solutions of nonlinear Hamiltonian systems

We wish to prove the existence of a family of periodic solutions of the N-dimensional dynamical autonomous system S N (6.4.1)

x,,

+ V U ( x )= 0

parametrized by the mean value of U ( x ( t ) ) over a period by suitably restricting the potential function U ( x ) . In this problem as in Liapunov’s criterion of Section 4.1, the period of a possible periodic solution is the implicit parameter in question. In contrast to the discussion of Section 4.1 in this case there is no obvious first approximation to the desired family of periodic solutions. Moreover, the obvious isoperimetric variational problem ( r R )of maximizing the potential energy of S N ,JAU(x(s))ds over the 1-periodic N-vector functions x ( t ) with fixed nonzero kinetic energy jA\.k12 ds = R and its dual, does not yield the desired family. Indeed, the maximum in ( r R )is easily seen to be infinite for “coercive” potential functions U ( x ) . While the minimum in the isoperimetric problem dual to (rR)is zero. However, on the basis of (6.3.4), we shall prove (6.4.2) Theorem Suppose that the function U ( x ) is a C” convex function defined on R N such that (i) O = U ( 0 ) 0 such that the mean value of U ( x , ( t ) ) over a period is R . Moreover if U ( x ) = A x . x higher order terms near x = 0 with A a positive definite matrix. then as R+O the period of x R ( r ) tends to the smallest nonzero period associated with the linearized system.

+

Proof: The result is proven by first establishing its validity for strictly convex C 2 potential fumtions U ( x ) using (6.3.4), and then for a general C ’ convex potential function U ( x ) satisfying (i) and (ii), by approximating U ( x ) with suitably chosen strictly convex functions U N ( x ) . Step 1: We first prove (6.4.2) in case U ( x ) is a C 2 strictly convex function U ( x ) satisfying (i) and (ii). To this end, we exhibit the period parameter explicitly by the change of variables t = XS in (6.4.1), and seek a 1-periodic solution x ( s ) ( E 0) of the system x,,~ X 2 V U ( x ) = 0 (6.4.3) since such solutions x ( s ) correspond to X-periodic solutions of (6.4.1).

+

338


Clearly, by (6.3.4), under our hypotheses such a solution can be found solving the minimization problem ( r N ) Minimize JAIxs12ds over the class of N-vector, absolutely continuous, and square integrable functions WN(O,1) described in (6.1.8) subject to the constraints

(6.4.4)

e~ = (x(s)

I ~ ( 3 )E W,,

I

U ( X ( S )ds ) = R,

I

l

V U ( X ( S )ds ) =0 .

Indeed, when regarded as an operator on W,(O, l), the kernel of x, consists of the constant N-vectors, so that the constraint in (6.3.4) coincides precisely with the constraint (6.4.4). Thus to complete Step 1, on the basis of (6.3.4), we need only note its hypothesis that clearly 97.(x) = J A U ( x ( s ) ) ds is a C2 strictly convex functional on W,(O, 1). The result (2.5.6) implies that 9l(x) is weakly sequentially continuous, while the coerciveness of ( L x , x) follows as in (6.3.9) from Jensen's inequality. Indeed, if x(s) = (x,(s), . . . , x,(s)),

eR

U ( i ' x l ( s )ds, . . . ,

0.

If X(s) is identically constant, say X(s) = c, then grad U ( c )= 0. Hence x . grad U ( X )= 0 if x is on the line segment joining x = O and x = c, so that U ( x ) = O on this segment. In particular, U ( c ) = 0, which contradicts the last formula. Thus X(s) z const. Furthermore, setting = xk in (6.4.5). and taking limits, we find (A,') is uniformly bounded (since x ' grad U ( x ) > 0 whenever U ( x ) > 0). Hence (A:) has a convergent subsequence with limit h2 # 0. Thus I satisfies

+

L'i .x'

ds = A*

L

I

+

grad U ( a ) . ds

for all

+ E W,.

Thus X is smooth and also satisfies x' + A 2 grad U(X)= 0 and is clearly the desired critical point of the isoperimetric problem. Now we show that ( j ~ ~ X , ( sds) ) ~ *is uniformly bounded, to complete the proof of the theorem. To this end, let x(s) = (sin 2ns, 0, . . . , 0). Then there exists a number r, > 0 and a vector ck E R" such that (by virtue of the properties of U,) yk(s) = t k x ( s ) + ck

sk, R )

that is, jo'grad Let C

Uk(Yk(X))

> 0 be so large that

ds = 0,

J,'Uk(YkW) = R ;

1x1 > C implies that U k ( x ) > 2R for k = 1, 2,

2R meas(s 10 < s < 1, lyk(s)l > C )

.. .

. Then

< R,

so that meas(s 10 G s

< 1, lyk(s)l
f

Thus there is an interval [{, q ] c [0, I ] on which Iyk(s)l < C, q - { > 4, and lsin 2q - sin 251 > 8 1 - sin t > 0. Since IyI(q) - yk(5)I < 2C, it follows that rk < 2C/6', and hence

Consequently, the sequence (I$Xk(s)12dr) is also bounded by 4 ( 2 ~ Z / 8 ) ~ . Remark: Theorem (6.4.2) can be considerably weakened if the potential function U ( x ) is even. Indeed, the convexity hypothesis can be replaced by the assumption that V U ( x ) .x > 0 for x # 0. This result is easily obtained by considering the closed subspace of W,(O, 1) consisting of those functions x(s) that vanish at 0 and 1 and are odd in s.

The importance of such global results is well illustrated by considering the preservation of periodic orbits of the well-known Kepler two-body problem under small autonomous Hamiltonian perturbations. More precisely, for N = 2 or 3 , in appropriate Cartesian coordinates, this problem can be written in the form (6.4.6)

where V ( x ) is a C' real-valued function with I V V(x)l = o(1) for 1x1 small, and e is a small parameter. We suppose that for e = 0, the system described by (6.4.6) has negative total

340


energy; i.e., 1x1’ - ( I / ( x l )< 0 for any solution x ( r ) . Then all the solutions of (6.4.6) with E = 0 will be periodic, and we seek those periodic orbits near x = 0 that are preserved for E # 0. Clearly an important problem to be surmounted is the behavior of the term ( x / l x l ’ ) near the singularity x = 0. To overcome this difficulty for N = 2, we apply the useful regularization theory of Levi-Civita, mentioned in Section 1.1, and show the importance of our global results on the periodic solutions of (6.4.6). To this end, suppose x = (xl, x 2 ) E R2 is written in complex notation x = x , + x 2 = u2 with u = u , + iu2 and r = 1x1. Also, make a change of independent variable s = jb(dt/r), so that d s / d r = I / r . Then on the fixed energy surface H = $ li 1’( I / r ) - E V = c, we find that ( l / r ) x ’ = x , so that restricted to this surface the Lagrangian becomes L’ = rL = (./2)(Ix’12/r2

+ 1+

f(C

+ EV)).

Since )x’12= 4 ( ~ ( ~ ) and u ’ I r~ = IuI2, in terms of u, L* becomes L = 2(u‘I2+ 1 Consequently, the transformed equations can be written as (6.4.7)

ii + V U(u)= 0,

where U ( u ) =

-

Iul2( c

+ uii(c + E V ) .

+ cV(u2))

Thus the periodic solutions of (6.4.7) with (x-(’ - ( I / ( x ( )- EV = c are in one-to-one correspondence with the periodic solutions of (6.4.7) such that (6.4.8)

lIi12

+ U(u)= 1

Clearly to find periodic solutions of (6.4.6) that satisfy (6.4.7) and (6.4.8), results of a global nature are necessary. Indeed, it does not suffice to know the existence of solutions of (6.4.7) near u = 0 since such solutions do not satisfy (6.4.8). Our result (6.4.2) yields a family uR(r)of periodic solutions of (6.4.7) for suitable functions V , one of which will satisfy (6.4.8).

6.48 Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds with vanishing Euler-PoincarB characteristic

Here we consider the problem of determining necessary and sufficient conditions on a given Holder continuous function K ( x ) that will ensure the solvability of the following semilinear elliptic partial differential equation defined on a compact two-dimensional Riemannian manifold (%, g) with x(%) = 0,

+

(6.4.9) Au - k(x) K ( x ) e 2 ”= 0, where A denotes the Laplace-Beltrami operator on (%,g) and k(x) is a smooth function with j,k(x) dVg = 0. As mentioned earlier, (6.2.9), a n affirmative solution of this problem has the following geometric meaning: K ( x ) is the Gaussian curvature of a Riemannian metric (%,g) with g = e2”g (pointwise) conformally equivalent’ to g. A more general notion of conformal mapping involves the composition of a pointwise conformed mapping with a diffeomorphism of % (see Fig. 6.1). From this geometric point of view, the immediate necessary condition for the solvability of In Kazdan and Warner (1974) the result (6.4.10) is used to prove the converse of the Gauss-Bonnet theorem for x(%) = 0.

6.4

ISOPERIMETRIC PROBLEMS IN GEOMETRY A N D PHYSICS

34 1

FIG. 6.1 The conformal mapping necessary to solve the general converse Gauss-Bonnet theorem for x(%) = 0.

(6.4.9), viz. 0 = J % K ( x ) e i u dV,, obtained by integrating (6.4.9) over %, means that the "integru curvuturu" of Em relative to the metric g must satisfy the Gauss-Bonnet theorem. This is precisely the kind of "natural" constraint we mentioned in Section 6.3B. On the other hand, from the point of view adopted in Section 6.3, the equation (6.4.1) occupies a singular position since the functional ( u ) = juR { 5 I V uI2 + k ( x ) u 2 } dV is clearly not coercive on the set S = { u I u E W , ,2(9Tl,g ) , j . a K ( x ) e 2 " dV = 0 ) . This implies that in order to represent a solution of (6.4.9) as a minimum. the set S must be supplemented. We shall show that we can regain coerciveness by the addition of a simple explicit constraint in the natural isoperimetric problem for (6.4.9). In fact, we shall prove the following sharp result. (6.4.1 0) Theorem Suppose the Euler-Poincare characteristic of 9, = 0. Then the equation (6.4.9) is solvable if and only if either K ( x ) = 0 or K ( x ) changes sign on % and j G m K ( x )eZUOdV < 0, where uo is any solution of Au = k ( x ) on % .

x(%)

Proof of necessity: If u satisfies (6.4.9) and

x(.%)

= 0,

then jLR K(x).

e2" dV = 0. Thus if K ( x ) is not identically zero, K ( x ) must change sign on

%. On the other hand, if we set u = uo + w , the function w satisfies the equation Aw

+ K(x)e2U0+2W = 0.

Multiplying this equation by e - 2 w ,integrating over Em, and integrating by parts, we find e-2w1

v wI2 c

i =~ -

K(x)e2'O d~

> 0.

342


Before proceeding further, we state an isoperimetric problem for the solutions of (6.4.9). Lemma Suppose ~ ( " n t=) 0 and K ( x ) is a given function defined on ;.?n such that relative to some Riemannian metric g defined on %, J w K ( x ) eZUOdV < 0. Then the (smooth) critical points of the functional @ ( u ) = ( t I V uI2 + k ( x ) u } dV subject to the constraint

1%

are (apart from a constant) solutions of the equation (6.3.9).

Proof: The proof of (3.1.31) shows that a smooth critical point u of the isoperimetric variational problem (defined above) satisfies

p o { A u - k(x)} + P I K ( x ) e 2 "= Pz, where pi ( i = 0, 1, 2) are constants (not all zero). Clearly, Po # 0 since otherwise u E S' implies PI = p2 = 0, and so we set Po = 1. Since J = K ( x ) . e2'0 dV # 0, no solution of Au - k ( x ) = 0 lies on S ' . Thus both PI and p2 cannot be zero. To show that p2 = 0, we integrate (j-) over and find

(t)

Since J q k ( x ) dV = 0, and u E S ' , P2 = 0. Since PI # 0, there is a constant c such that k e2' = PI. So P = u c satisfies AP - k ( ~_t) K(x)e2' = 0. Now we show the PI > 0, so that PI = eZC,and consequently P = u c satisfies equation (6.4.9). Set u = uo + w in (t).Then by hypothesis, since p2 = 0,

+

Aw

+

+ PIK(x)e2U%2W = 0.

Again multiplying by e - 2 w , integrating over 9 2 ,and integrating by parts, we find

Thus PI > 0 since w # 0.

Proof of sufficiency: To prove the existence of a critical point for this variational problem, we set u = uo c , where Jwuo dV = 0 so that

+

6.4 ISOPERIMETRIC PROBLEMS I N G E O M E T R Y A N D PHYSICS

343

Consequently, s3 (a,,) is coercive on S', and 4 (a) is weakly lower semicontinuous. Furthermore, S' is closed with respect to weak convergence on W , ,2( %, g), and so by (6.1.l), inf 4 (a) over S' is attained by an element u E S'. Thus u is a weak solution of the equation (6.4.9) in the space Wl, 2( W ,g). Since u is a weak solution of a linear equation of the form Au = f with f E Lp for all finite p > 1 , u is smooth enough to satisfy equation (6.4.9) in the classical sense, and the theorem is thereby proved. 6.4C Riemannian manifolds with prescribed scalar curvature

Let (0311"', g) be a given compact Riemannian manifold of dimension N > 2. On such a manifold we seek a new metric defined on 3RN of the form = e2"g such that the new Riemannian manifold ('131tN, g) has prescribed curvature g(x) < 0 on 9 R N .As mentioned in Section 1.1A, if k ( x ) denotes the scalar curvature of ( 9 R N ,g), then the partial differential equation defining u can be written in the form

(6.4.1 1)

4(N - I ) N-2

+

Ali

+ g ( x ) u " = 0, 2) and u = exp 4 ( N 2)u must be strictly posi-

- k(x)u

where u = ( N 2)/(N tive on 9 L N . Here A is the Laplace-Beltrami operator relative to ( "XN, 8). From our discussion of (6.3.8), it follows that special properties of the differential geometric problem must be used to solve (6.4.1 1) since the exponent u is the critical value relative to (6.3.8). Actually we shall prove

(6.4.12) Suppose j C m ~ k ( dV x ) < 0, then (6.4.1I ) has a strictly positive smooth solution u ( x ) defined on ( : X N8). , Consequently EmN admits a Riemannian metric S (pointwise conformally equivalent to g) with prescribed scalar curvature g(x) < 0. Proof: By repeating the argument given in (6.3.8), we can find a positive smooth solution (uf, A,) of the system

for any small c > 0. Furthermore A, < 0 since A, is the minimum of the functional @ ( u ) = /%.N(2(N - 1 ) / ( N - 2)1 V uI2 + k ( x ) u 2 ) dV over the functions in w,,,(%",g) with ?Bf(u) = I % N I g ( x ) l u a + I - f d V = 1. For if c is the positive constant such that = C 2 / , N k ( X ) dV 0.

0}, we find that $ must satisfy

On the unknown boundary dA of the vortex ring, we suppose (i) grad 4 is continuous, (ii) $ = 0; while on the axis of symmetry r = 0 we set rc/ = - k

6.4 ISOPERIMETRIC PROBLEMS IN GEOMETRY AND PHYSICS

349

‘t

FIG. 6.2 Notation and expected streamline pattern for a steady vortex ring.

< 0, where k is a prescribed flux constant. Finally, we suppose that the velocity field of the ring tends to a constant vector (0, W , 0) at infinity. This fact can be obtained by requiring (6.4.24)

$

+ 4 Wr2+ k + O

as r 2

+ z 2-+

00.

([I) Reformulations The free boundary value problem formulated in (i) is difficult to treat both because of the unknown domain A and because of the nonlinearity of (6.4.23). To separate these difficulties, we reformulate the problem as a semilinear Dirichlet problem on TI, which is untroubled by the unknown domain A . To this end, we extend the function f(t) to the entire real axis by setting f(t) = 0 for t < 0, a n d \k = $ - t Wr2 - k . Then the desired vortex ring can be found by solving (6.4.25)

1

$,r

-

r

$r

+

$zz

-

by($

-

Wr2 - k ) = 0

on TI,

$ I a n = 0. The maximum principle for second order linear elliptic equations will imply that the cross section A of the vortex ring can be found from a solution of (6.4.25)-(6.4.26) by setting

(6.4.26)

(6.4.27)

A,

= { ( r ,z )

I $ ( r , z ) > + Wr2- k } .

This reformulation then requires us to find a nontrivial solution for the system (6.4.26)-(6.4.27) on the unbounded domain II,and moreover, unless

350


f(s)+O as s+O, the system has the added complication that the extended function f(s) may be discontinuous at s = 0. Fortunately, these two difficulties can both be overcome by limiting arguments': The domain II can be approximated by large rectangles n(a,6) with vertices (a, -+ b), (0, -+ b), with a, b large, and a function f(s) discontinuous at s = 0 can easily be approximated with a Lipschitz continuous one. Consequently it suffices to solve the system (6.4.25)-(6.4.26) with II replaced by II(a, b), assuming f is Lipschitz continuous, and then to investigate the set A,. (111) Resolutlon of the problem (6.4.25)-(6.4.26) on n ( a , b) We begin by regarding the system (6.4.25)-(6.4.26) as a gradient operator equation on an appropriate Hilbert space H . Here the space H can be conveniently chosen as the closure of C?(JJ(a, b)) relative to the inner product

where dr = r dr dz. Relative to this inner product the appropriate generalized solution of the adjusted system (6.4.25) can be written conveniently as ( u , +) =A//

yf(\k)+ dr

for all

n(a,b

+E H.

Now we are in a position to prove the following with D

=

n(a, b):

(6.4.28) Theorem Suppose f(t ) is a Lipschitz continuous nondecreasing function defined on [0, 00) with f(0) = 0 and with polynomial growth. Set F ( s ) = jif(t)dt. Then the system (6.4.25) has a smooth solution #(a, b) on n(a,b), for each k > 0. Moreover, #(a, b) has the following properties: $(a, 6) is even in z , strictly positive in n(a,b), and is an extremal of the functional J ( u ) = F(u - 4 Wr2 - k)r dr dz over llullH = 1. Also, if f E C ' and is convex, then the set A is simply connected. Proof: We first note that Hilbert space H can be continuously imbedded in the standard Sobolev space W,,2(II(u,b)), so that the imbedding theorem of Section 1.4 applies equally well to H or to k , , 2 ( I I ( u ,6)). To this end, we simply note that for u E H , (u:

+ u,') dr dz
0.

1

(This would not be true for the analogous one-dimensional problem.) Since F ( t ) is strictly increasing, it suffices to observe that functions with small norm in H can have arbitrarily large values on a small set. For example, given a point xo E n(a,b), the function

I

0

has Ilup(x)ll = 1 for a certain value Ix - xoI

+o.

for

Ix - xoI > 6,

p sufficiently large, but tends to cc as

Clearly the functional J ( u ) is continuous with respect to weak convergence in H , by the Sobolev imbedding theorem, and the polynomial growth condition on f (u). Indeed,

where L,(D, T ) denotes the Lp functions over n(a,b) with the volume element d7 = r dr dz. Now the argument of (6.3.1) shows that p is attained by an element $ E H . Moreover, $ > 0 in H ( u , b ) since the nonnegative part of $, $+, has the property that .I($+) = J($), while [I$+ 11 < II$II if $ < 0 on a set of positive measure. Thus a scaling of $+ would be the desired extremal element. By virtue of (3.1.31), we therefore find that there are constants p , and p2 (not both zero) such that (6.4.30)

for all w E H .

To show that pl = 0 and X = p J p I > 0, it suffices to set w = $ in (6.4.30) and to observe that then the integral on the left-hand side of (6.4.30) is positive. Since F ( t ) < t f ( t ) , 0 < *(x) < +(x) for $(x) > 0, and we have

The fact that $(a, b) is even in z follows from Steiner symmetrization with respect to the line z = 0 in n(a,6). The regularity of n ( a , 6) follows by standard regularity theory despite the apparent singularity of the coefficients of the equation at r = 0. To complete the proof of the theorem, we show that the set A, defined by (6.4.27) is simply connected under the convexity hypothesis off. To this

352


end, we consider the second variation formula of (3.1.40) for arbitrary u satisfying (u, 4) = 0

We suppose that A , has at least two components E l , E,. Let the functions w I and w 2 be defined by setting wi = 9,on E, ( i = 1, 2 ) and zero elsewhere. = Then if u = c l w l + c2w2, where cl, c, are chosen so that - c 2 ( ( w 2 ( ( ’ we , find that, from the above, contrary to (6.4.29),

>0

(by the convexity off).

(Iv) The llmltlng procedure as n(a,b)+II We now show that a limiting procedure applied to the results of (iii) for II (a. b ) solves the problem (6.4.25)-(6.4.26) as formulated in To this end we let the rectangles II (a. b ) tend to the half-plane II by letting (a, b)+ to. The results of (iii) then yield solutions ( + ( a , b), A(a, b)) with A(a, 6 ) contained in a bounded set of real numbers (uniformly bounded above zero) of the modified problem as stated in = +(a,,, b,) in H ( n ) = W , ,,(n) Theorem (6.4.28). We seek a convergent subsequence obtained by setting $(a,,, b,,) = 0 outside II(a,, b,) with associated convergent eigenvalues A(u,, b,) = A,. To this end we shall prove the existence on a fixed domain Sl such that the associated vortex cores A ( + ( a , b)) c Sl no matter how large a and b become. This domain 51 will have the property that vol[Sl n ( y 1 I x - y J < I)]+O as J x J + t o so that the imbedding H(Q)+ L,(Sl) is compact, and consequently one can pass to the limit in the integral equation

n.

+,,

( ~ . yis)Green’s function on n(a,b ) associated with the linear operator of (6.4.25). To find the domain Q we first prove the following a priori bound for the solution of (6.4.25)-(6.4.26) on n(a,b). Go,b

Lemma Let I(u) be the length of the projection of the set A ( u ) = ( ( r , z ) 1 ( r , t) E n(a, b), u ( r , t) > ) r 2 + k ) on the

I

axis, where u

c C’(II(u, b)) and

Y ( u ) = { ( r , z ) I ( P , I), A (u ). P

> r > 0)

(i.e., Y(u)is a set in n ( a , b) containing A ( u ) and all points ( r , z) in point ( p , z ) of A ( u ) and the z-axis). Then

(6.4.32)

JJy(u{

dr dz -I-2kl(u)

n(a,b) that lie between a

IIUIIL(,.by

Proot: Let f r 2 = y and assume the boundary of Y(u) is smooth so that the divergence theorem can be applied there. Then a simple computation, using the divergence theorem, shows

6.5

CRITICAL POINT THEORY OF MARSTON MORSE IN HILBERT SPACE

353

Since u = 0 on the z axis, and on 3A ( u ) , u = y + k , we find from the last set of inequalities the desired bound (6.4.32). We find, applying the lemma just obtained to u = +(u, b) that the set 0 can be chosen as Q = ((r, z) I It(< ( r 2 + 4 k ) - ’ ) . Indeed since /l+(u, b)llH = 1 and by Steiner symmetrization the projection of A ( + ) on the z axis must contain the interval of the form IR = ( - h ( R ) , h ( R ) ) , so that the open rectangle (0, r ) X I, must be contained in Y(u). Then applying (6.4.32) and noting that 2 h ( r ) d /(+) we find f r2(2h(r)) + (2k)(2h(r)) < 1, whence h ( r ) < ( r 2 + 4 k ) - ’ . Consequently we can find a weakly convergent subsequence (I)“, A,) in H ( I I ) , and pass to the limit in (6.4.25), as mentioned earlier. The weak limit (4, A) then yields the desired nontrivial solution

(v) Historical background In 1858, Helmholtz considered vortex rings of very small cross section as one of two examples of his theory. Basing himself on Helmholtz’s results on the indestructibility of vortex rings and taking the “ether” as the appropriate ideal fluid, Kelvin made vortex rings the basis of a primitive atomic theory. Kelvin conjectured the existence of nonaxisymmetric vortex rings whose “core” could be associated with thickened knot configurations in R (The differing topological structures of knots in R3 were supposed to classify the various atomic structures in Nature.) Indeed, on this basis, Kelvin’s collaborator, Tait, was thus led to pioneering work in the mathematical theory of knots. Kelvin’s theory declined with the overthrow of the “ether” concept. Nonetheless, in modern theoretical low-temperature physics the importance of vortex rings has strikingly reappeared in the theory of superfluidity and superconductivity since there ideal fluids approximate reality.

’.

6.5 Critical Point Theory of Marston Morse in Hilbert Space

In order to study all the critical points of a given functional 4(u), defined on a Hilbert space H , topological considerations are necessary to supplement our earlier discussion. This point has been clarified in both finite- and infinite-dimensional contexts by Marston Morse beginning with studies dating from 1925. To illustrate this point, we note that for a linear compact self-adjoint operator C E B( H , H ) , orthogonality notions can be used to develop a complete spectral theory once the analogue of (1.3.40(i)) is established. Moreover the principle of superposition for linear operators implies that for a self-adjoint Fredholm operator L , the solutions of the operator equation Lu = f fill up finite-dimensional linear subspaces of H . In a nonlinear context, totally new ideas must be used to find analogues of these results. In this section we shall describe Morse’s approach to this problem in a simple Hilbert space context. In Section 6.6 we describe the related theory of Ljusternik and Schnirelmann, while in Section 6.7 we illustrate the use of these theories in various contexts.

354


6.5A A sharpening of the steepest descent method

For any general study of critical points, it is necessary to supplement the notions of weak lower semicontinuity and coerciveness by more refined analytic and topological considerations. To this end, we first reconsider the method of steepest descent, introduced in Section 3.2. Suppose F ( x ) is a smooth C 2 functional defined on a real Hilbert space H , and bounded from below on H . Then in Section 3.2 it was shown that the solutions x ( t , xo) of the initial value problem (6.5.1)

dx/dt

= -F’(x),

~ ( 0x0) , = x0,

exist for all t 2 0, with lim x ( t , xo) as t+ co a critical point of F ( x ) provided that the critical points of F ( x ) are isolated and that F ( x ) satisfies the following compactness condition (mentioned earlier in (6.1.1’)). (6.5.2) Condition (C) Any sequence {x,,} in H with IF(x,,)I bounded and \\F’(x,,)\l+O has a convergent subsequence. Under hypothesis (6.2.2), the discussion of Section 3.2 shows that F ( x ) clearly attains its infimum on H . The following result shows the utility of (6.5.2) for the study of other types of critical points. (6.5.3) Theorem Suppose that a C 2 functional F ( x ) defined on H is bounded from below and satisfies (6.5.2), and has only isolated critical points. If F ( x ) possesses two isolated relative minima yI,y2, then the functional F ( x ) must possess a third critical point y3, distinct from y , and y 2 , which is not an isolated relative minimum. Proof: Suppose that F ( x ) does not have a third critical point. Then we shall show that H can be represented as the union of two open, disjoint subsets U , and U,; which obviously contradicts the connectedness of H . To construct the sets U j , suppose that x ( t , xo) is the solution of (6.5.1). By (6.5.2), x ( t , xo) exists for all t 2 0 and lim x ( t , xo) as t + co is yi ( i = 1, 2). Let Ui = {xo 1 lim x ( t , xo) = yi as t + co} ( i = 1, 2). Clearly H = U , u U,, while U , and U , are disjoint. To show that the sets Uj are open in H , we first note that each y j ,being a strict relative minimum, has a neighborhood W j such that any solution x ( t , xo) which enters Wj remains in Wj and, in fact, converges to yi as t -+ w . Indeed, for xo sufficiently near yi, since F ( x ( t , x,)) is a decreasing function of t , x ( t , xo)+yi. Thus, by virtue of the continuity of x ( t , xo) with respect to the initial condition xo, if zo E U,, then for c > 0 sufficiently small with llz, - Zoll < E , there is a T such that both x ( T , t o )and x( T, 2,) lie in W j .Consequently Z, E U, if zo is. Therefore each Ui is open, and we have obtained the desired contradiction. It is also immediate that the third critical point y 3 shown to exist by the

6.5 CRITICAL POINT THEORY OF MARSTON MORSE IN HILBERT SPACE

355

above argument cannot be another relative minimum, for if it were and if F ( x ) had no other critical points, then the argument just given would again lead to a-contradiction. 6.5B Degenerate and nondegenerate critical points

To proceed with a deeper study of the critical points of a C 2 functional F ( x ) defined on H , the following definitions are convenient. A critical point xo is called nondegenerate if the self-adjoint operator F ” ( x o ) is invertible. Otherwise xo is termed degenerate. The index of a critical point xo of F ( x ) is the maximal dimension on which the form ( F ” ( x , ) x , x) is negative definite. F ( x ) is a Fredholm functional if F ’ ( x ) is Fredholm. The nondegenerate critical points of F ( x ) have several important properties. First, by the inverse function theorem (3.1S) , the nondegenerate critical points of F ( x ) are isolated. Next, the set of C2 functionals F ( x ) defined on a bounded subset of H such that the critical points of F ( x ) are all nondegenerate form a dense subset of all such C 2 Fredholm functionals. Indeed, if G(x) is a C 2 Fredholm functional _some of whose critical points may be degenerate, consider th_e functional G,(x) = G(x) ( x , p ) . Clearly, for llpll sufficiently small, IIG,(x) - G(x)llc2 can be made as small as desired. On the other hand, by (3.1.5), all the critical points of 6,(x) are nondegenerate providedp is not a singular value for G’(x). Since C’(x) is a C ’ Fredholm operator of index zero, by (3.1.45), this set is nowhere dense in H . Furthermore, if a C2 functional F ( x ) has all its critical points nondegenerate on F - ‘ [ a ,b ] and F ( x ) satisfies Condition (C) of (6.5.2), then there can be at most a finite number of critical points on F - ’ [ a , b ] for any - 00 < a, b < 00. Indeed, otherwise there would be a ~ \ so that { x n > would sequence { x n > with a < F ( x , ) Q b and ~ ~ F ’ ( x ,=, )0, have a convergent subsequence with limit X. This would be a contradiction, for clearly X would be both a nonisolated and nondegenerate critical point of F ( x ) with X E F - ’ [ a , b ] . Another interesting result is the following extension of Morse’s lemma (1.6.1) to a Hilbert space context. (6.5.4) Theorem Suppose F ( x ) is a C 3 functional defined in the neighborhood of a nondegenerate critical point x = 0. Then there is a diffeomorphism h mapping a neighborhood U of x = 0 onto itself such that for x E U , F ( x ) - F(0) = (F”(O)h(x),h ( x ) ) .

+

Proof: Since x

=0

is a critical point, we have I

F ( x ) - F(0) =

( ( F ’ ( s x ) ,x) ds,

JO

356


while

1( d / d t ) F ‘ ( t x )dt 1

F‘(x)=

=

0

Thus we can represent F ( x ) in terms of F“ by writing I

F ( x ) - F(0) =

j’j s ( F ” ( s t x ) x ,x ) dr ds 0

=

( k ( x ) x ,x ) (say),

0

where we may suppose that k ( x ) is the self-adjoint operator defined by setting ( k ( x ) y , z ) = j ’ j ’ s ( F ” ( s t x ) y ,z ) dt ds. 0

0

Clearly with this definition, 4 (F”(O)x,x ) = (k(O)x, x ) . Now suppose that B ( x ) = [ k ( O ) ] - ’ k ( x )and C ( x ) which exists for llxll sufficiently small, since B(0) = I (the identity). Also note that, if C T ( x ) denotes the adjoint to C ( x ) , then C T ( x ) k ( 0 )= k(O)C(x) since B T ( x ) k ( 0 )= k(O)B(x), and

=vm

[ k - ’ ( 0 ) C T ( x ) k ( 0 ) 1 2 =k(0)-’BTk(O) = B ( x ) = C * ( X ) .

Thus the result is obtained by the following simple computation: F ( x ) - F ( 0 ) = ( ~ ( x ) xX ,)

= ( k ( 0 ) C 2 ( x ) x X, )

= ( C T k ( O ) C ( x ) x ,x ) = ( k ( O ) h ( x ) ,h ( x ) ) =

+

( F ” ( O ) h ( x ) ,h ( X ) ) >

where h ( x ) = C ( x ) x is invertible for llxll sufficiently small. This result has the immediate consequence that after a local differentiable change of coordinatesy = Y ( x ) near the nondegenerate critical point x = 0 the functional F ( x ) can be written in the form

F ( Y ) = IN1 - P)Yll2 - IIPYI127 where P is the projection of H onto the linear subspace of H on which F”(0) is negative definite. We next observe that exactly as in the finite-dimensional case our discussion applies when the Hilbert space H is replaced by an infinitedimensional manifold % that locally approximates H . This fact is very useful for many differential geometric problems (see the notes at the end of the chapter). More explicitly, Deflnltlon A manifold G3n, of class C‘ modeled on a Hilbert space X (is., Hilbert manifold) is a collection of open sets { U,} and mappings 0,: U, X such that:

6.5

CRITICAL POINT THEORY OF MARSTON MORSE IN HlLBERT SPACE

357

(i) 6,: U, + O,( U,) is a homeomorphism, n U,) is a smooth map of class C'. (ii) O a 6 i 1 : O,(U, n U,)-6,(U, Definition A n atlas of class C' for a set 9,is a collection of open subsets { U,} of X . and transition maps such that:

(i) sb, is a homeomorphism of U, onto {,(U,), a subset of a Hilbert space X ; (ii) the mappings are compatible in case U , n U p # 0 , i.e., { J i ' : la( U, n U,)+ {,( U, n U p ) is a homeomorphism of class C' for each a,P ;

(iii) the collection (U,, and (ii).

5,)

is maximal with respect to properties (i)

A Banach manifold of class C' modeled on X is an atlas of class r defined on the set Lm relative to the Banach space X . The members of the collection {( U,, la)} are referred to as charfs.

Most properties of a mapping f between spaces, defined by means of the derivative o f f , can be extended to mappings between Banach manifolds via this definition. For example, Definition Let f: 9,4 % be a mapping defined on the manifolds % and %. Then f is of class CP, provided it is continuous and relative to charts at each x E .?Tl and f(x) E ?R ,f is CP; i.e., { , h f O i l is a smooth map of class CP between the Banach spaces X , and X,. These definitions enable us to carry over much of the local analysis of Chapter 3 to the study of mappings between Hilbert manifolds. In particular a tangent space, T a x to a Hilbert manifold "L at x E 3R is the set of all tangent vectors (~'(0))to 3R, relative to C ' curves p ( t ) passing through x at t = 0. Moreover, the differential of a mapping f: '32 + 9'L between two Hilbert spaces is the mapping df(x): T ' X , T9L,(x)defined by setting

-

d

df(x)(p'(O)) = ;i;f ( P ( 0 )

I 1-0

for every curve p(r) and so is linear in the argument p'(0). For a smooth functional 4 ( x ) defined on a Hilbert manifold " X ,the differential d 4 (x,p'(O)) is linear in the argumentp'(0) and thus we write d 9 ( x , y ) =(grad 9 ( x ) , y )

by virtue of the Riesz representation theorem for linear functionals. Furthermore, in terms of charts about x and f ( x ) , the differential df(p'(0)) can be computed as a derivative of a mapping. The critical points of a functional 4 ( u ) defined on a Hilbert manifold % thus coincide with the points x E "X such that grad 4 (x) = 0. Clearly notions such as critical point, degenerate and nondegenerate critical point, and Morse index, being invariant under coordinate changes, have a well-defined meaning on '9-. Many of the results we obtain in the remainder of this section can be adapted to apply to Hilbert manifolds. In order to investigate the relations between the critical points of F(x), we use singular homology. The basic properties of singular homology groups with coefficient group 9 can be summarized as follows:

358


(i) Iff: ( X , A ) + ( Y , B ) is a continuous mapping, then there is a group homomorphism for each integer q,f.: H , ( X , A ; '3)+ Hq( Y , B ; 9) with the following properties. (a) I f f = i , the identity i. is the identity automorphism. (b) If g: ( Y , B ) - + ( Z , C ) , then (gf). = g.f.. (c) df. = f.d. (d) (Hornotopy property) If& g: ( X , A ) + ( Y , B ) are hornotopic, thenf. = g.. (ii) (Excision property) If U is an open subset of X with ( X - 17, A - ( I ) + ( X , A ) induces the isomorphism

e.: H q ( X

-

U,A

-

U ;9)-+ H,(X,A ; 9 ')

a c int A , the inclusion e :

(for each q).

(iii) (Exactness property) If i : A + X and j : X +(X, A ) denote inclusion maps, then the following infinite sequence is exact, i.e., the image of each homomorphism equals the kernel of the next homomorphism

(iv) (Dimension property) If X is a space consisting of one point, then

H,(X; G ) =

0

9

for for

p # 0, p = 0.

6.5C Morse type numbers In order to investigate the structure of all the critical points of a given functional, it is useful to establish a classification of critical points that is invariant under local differentiable changes of coordinates. The following classification (due to M. Morse) will be used in this section. First, consider the nondegenerate critical points of a smooth functional F(x). We classify these by their index. The result (6.5.4) ensures that such a classification is invariant under local diffeomorphism. Assuming F ( x ) possesses an isolated degenerate critical point xo, we associate with xo a sequence of positive integers (Mo(xo),M , ( x o ) , M2(x,,), . . . ) called the type numbers of xo. The integers Mi(xo), i = 0, 1, . . . , are a measure of the number of nondegenerate critical points of index i equivalent to xo. These type numbers M i ( x o ) are defined as the Betti numbers of the relative singular homology groups Hi(FC+' n U , Fc-' n U ) with 2, coefficients, where F(xo) = c, F d = ( x F ( x ) 2 d } , and U is a small neighborhood of xo (c > 0 is sufficiently small). In order to justify these definitions, we prove

I

(6.5.5) Theorem Suppose F(x) is a C 2 real-valued functional defined on a Hilbert space, which satisfies condition (6.5.2), and suppose b > a.

(i) If F ( x ) has no critical points on the interval [ a , 61, then the sets F b = {x 1 F ( x ) < b } and F a = ( x I F ( x ) < a } are isotopic. Furthermore, the isotopy may be so chosen that the points of Fa are fixed, so that Fa is a deformation retract of F b .


359

(ii) If F ( x ) = c is an isolated critical value of F ( x ) on which F has only a finite number of critical points a(c) = { x i } ,then for any c > 0

H ~ ( F ~ +F' , - ~=)H ~ ( + n

uu

u(c),

Pc n u),

where PC= { x I F ( x ) < c} and U is any sufficiently small neighborhood of all the critical points u(c). (iii) If F ( x ) has a single nondegenerate critical point of index i on [ a , b],H q ( F b ,F") = 0, q # i, while H i ( F b , F") = G (the coefficient group of the homology theory). Proof: (i): First we observe that since F ( x ) satisfies Condition (C) for some co > 0 sufficiently small, F ( x ) has no critical points on F - ' [ a co, b + cO], and in fact there is a positive constant d > 0 such that infF-II,-,o,b+c,lIIF'(x)ll > d. Otherwise, there would be a sequence {x,} E H with x, E F - ' [ u - (l/n), b + (l/n)] and F'(x,)+O so that after passing to a subsequence x, + X E F - ' [ a , b],F'(X) = 0. Now to define the isotopy S;(x) of F b onto F a , we use the method of steepest descent discussed in Section 3.2, and consider the truncated analogue of (6.5.1),

(6.5.6)

dx = - a ( l F ( ~ ) l ) F ' ( x ) dt

IIF'(x)Il2

,

x(0) = xo E F - ' [ a , b ] ,

where a(.) is a real-valued C m nonnegative function such that a ( z ) = b a for a < z b co. Since

+

infI,-,o, bl IIF'(x)ll > d > 0, the right-hand side of the differential equation (6.5.6) is locally Lipschitz continuous and uniformly bounded. Furthermore, a simple argument using (3.1.27) shows that x ( t , xo) exists for all t E (- 00, 00). Now

Consequently, x ( t , xo) leaves F"-'o fixed (pointwise) and deforms Fb onto F a . Thus setting S,(xo) = x ( t , xo) for xo E F - ' [ a , b], we see that 5; is a deformation of F b onto F". In fact, 5, is an isotopy of H onto itself since for any xo E H with {-,(xo) = x( - t , xo), S-,S,(x,)

= L , ( X ( t , xo)) = x ( - 2 , x ( t , xo)) = xg.

Actually, we can prove (by a slight modification of the above argument) that F b is a deformation retract of F". Indeed, denote by x ( t , xo), the solution of (6.5.7)

dx = - ( F ( x O ) - U )

dt

F'(x) IIF'(x)112

, x(0) = xo E F - ' [ a , b].

360


Then the mapping l l ( x 0 ) = x ( t , xo) for xo E F - ' [ a , 61 and l l ( x 0 ) = xo for xo E F ( x ) < a. As before for xo E F - ' [ a , b ] , F ( S , ( x , ) ) = F ( x o ) t { F(xo) - a ) . Thus F(l,(xo))= a and F b is a deformation retract of F". (ii): Now suppose that F ( x ) = c is an isolated critical level of F on which F has a finite number of critical points {zi}( i = 1, . . . , N ) . We will show that F' is a deformation retract of FC+' for some small E > 0, by using the deformation l l ( x 0 ) defined by (6.5.7). In particular, we show that limlTlx ( f , xo) exists for xo E F - ' ( c , c €1. First, suppose (for i = 1, . . . , N ) , inflElo,,) ( I x ( t ,xo) - { z i } l l > 0, so that by Condition (C), llF'(x(t, xo))ll is uniformly bounded above zero. Thus for any two values 0 < t , , t2 < 1,

+

< K J F ( x o )- c where K is a constant independent of f . For any sequence i,,Tl, { x ( t n ,xo)) is a Cauchy sequence and limlT,x(t,xo) exists. Next, we suppose infIEI,,,) I l x ( t , xo) - { z l ) l l= 0, so that for some sequence t,tl and some integer i, Ilx(t,,, xo) - z,(J-0. Actually we prove limlT,x ( t , xo) = z I , by supposing the contrary, and obtaining a contradiction. Indeed, if this limit does not exist, there are two spherical neighborhoods S , c S2 of z, such that for an infinite sequence of disjoint intervals [t,, $ + , I c [0, 1) with x ( t , xo) C S2 - S , for f E It,, t,+ ,I, there are absolute positive numbers c, d > 0 such that (6.5.8)

llx($+,) - x(t,)\l 2 c

and

l\F'(x(f))ll 2 d.

inf [$, ',+

I1

Then (6.5.8) implies

This is the desired contradiction since asj-oo, excision H J F ~ + F~ ~, - ' )= H J +

I$+,

- $J+O.

Finally, by

u u n W , kcn w),

where W is any neighborhood of u that contains no other critical points of F(x)* (iii): First, suppose F ( x ) is a quadratic functional and x = 0 is the critical point of F ( x ) of index i with F(0) = 0. Then there is a self-adjoint, invertible operator L of H into itself such that F ( x ) = ( L x , x). Clearly Condition (C) implies that L is compact, so that L has separable range. Thus, we may suppose without loss of generality that H is separable. Since F ( x ) is negative definite on a closed subspace H , of dimension i of H ,

+


361

H = H , @ H;', and L is invariant on H , and Hi'. Therefore, if { x I F ( x ) < 01,

ko=

6.5

(6.5.9)

H q ( H ' , H - t ) = H q ( F ou {0}, FO).

Let Z be the open unit ball in H and set 2, = Z n H i . We shall show that (Xiu (0}, Z,) is a deformation retract of (h0u (O}, @. Once this is established, (6.5.5(ii)) implies that H q ( F C ,F - ' ) = H H q ( B ,U (01, B ; ) = H q ( B ; , B; - { O } ) = H q ( B ; , Mi).

To show that ( k ou ( 0 ) . ko)can be deformed into (C, u {0}, 2,). suppose that F ( x ) < 0 and x = .x,+ y . where x, E H , and y E H j L , and set x ( t ) = x, ( 1 - t)y. Then

+

F ( x ( t ) )= 4 ( W t ) , x ( t ) ) = 4 (Lx,, X I )

+ f(1

-t*)(L~,y)~~(Lx,x) 1, M , - M,> -1, M , - M , + Ma > 1, m

( - l)'M, = I. 1x0

362


where M idenotes the number of critical points of Morse index i of F ( x ) . Proof: This proof makes crucial use of the result (6.5.5) and the basic properties of singular homology theory. We first observe that the nondegeneracy hypothesis (iii) of F ( x ) and the compactness condition (6.5.2) imply that for any real number 6 , the number of critical points of F ( x ) on F b is finite. This fact follows from the isolated character of nondegenerate critical points since an infinite number of nondegenerate critical points of F b would be inconsistent whenever condition (6.5.2) is assumed. Then by our discussion of singular homology given above, as in Milnor (1963), for any subadditive integer invariant S,(X, Y ) n

S,(Fan,

P o )

0, whereas R o ( H ) = 1 . The result (6.5.5) has a useful extension if F ( x ) is defined only on a bounded domain of H , sayZ, = { x I llxll < R } . In fact, we prove (6.5.1 1) Corollary Suppose hypotheses (i)-(iii) of (6.5.5) are valid for the set Z, = { x I llxll < R ) . Then if ( F ' ( x ) , x) > 0 for each x on llxll = R , and Mi denotes the number of critical points of F ( x ) of index i in Z,, the inequalities (6.5.10) hold. Proof: We note that the condition (F'(x), x) > 0 implies that the solutions x(t,x,,) of the equations of steepest descent used in the proof of

6.5


363

(6.5.10) remain in the set u, for all t . Since the homology groups of Z, coincide with those of H , the proof given in (6.5.10) carries over to this case. In this case hypothesis (iv) of (6.5.10) can be removed since by hypothesis there are no critical points of F ( x ) on llx1/2= R , and in Z, the number of critical points of F ( x ) is finite. Finally, we consider an extension of the inequalities (6.5.10) for functionals F ( x ) that may possess degenerate critical points. To this end it is since convenient to consider functionals defined on Hilbert manifolds 9, such sets often possess nontrivial topological properties. In such cases any smooth functional defined on % that satisfies (6.5.2) will be shown to have an interesting critical point theory. Before stating a result in this direction, we define an integer-valued measure of the critical points on an isolated critical value F ( x ) = c, by setting M , ( c ) equal to the Betti number of the relative homology groups with Z , coefficients, H , ( F C + ' ,FC-'), for E > 0 sufficiently small, where as usual, Fd = { x I x E Ti,F ( x ) < d ) . Clearly as in (6.5.10), c . ( c ) is independent of E if F ( x ) satisfies condition (6.5.2) on the set { x I c - E < F ( x )

< c +r}.

We now state the following result that can be proved on the basis of (6.5.5(ii)). Note that it does not require the nondegeneracy of the critical points of F ( x ) . (6.5.12) Suppose F ( x ) is a C2 real-valued functional on a complete, smooth, Hilbert manifold 9R such that F ( x ) is bounded from below, satisfies condition (6.5.2), and possesses isolated critical values c,, c2, . . . . Then the following inequalities hold

M,(c,) 2 Ri(%),

(6.5.13) (c,)

where Rj is the ith Betti number of %. This result can be proved by a simple modification of our previous results. See Rothe (1973). 6.5E Illustrations

To illustrate the results just given, consider the simple semilinear Dirichlet problem (6.1.30) discussed earlier, defined over a bounded domain 0 c RN, (6.5.14)

c2 A U

+ u - g ( x ) u 3 = 0,

ulaQ= 0. Clearly the trivial solution uo(x)= 0 exists for all

E,

and its index is easily

364


computed to be the sum of the multiplicities of all eigenvalues of the Laplacian A relative to (subject to the null boundary condition) less than 1/c2. Indeed, the solutions of (6.5.14) are precisely the critical points of the c21 V uI2 - u2 g(x)u4> dx in the class k,, ,(SL), functional 4, ( u ) = JQ{ and ( gC"(O)u, u ) = J a { E'I~V uI2 - u 2 } d x . We also note that 4, ( u ) is bounded from below on W , ,,(a)and (since 4:(x) is a proper mapping for fixed c) satisfies (6.5.2). Let the eigenvalues of (A, 8) be denoted 0 < A, < A, < * . * < A, < * , so that uo(x) = 0 is a nondegenerate critical point of 4, ( u ) if and only if c - ' # Xi ( i = 1, . . . ). Then, as mentioned in (6.1.31) for l / c 2 < A,, uo(x) = 0 is an absolute minimum of 4 , ( u ) and the only solution of (6.5.14). However, for l/c2 > A,, u, is no longer a relative minimum. As proven in (6.1.31) for A, < 1/c2, the infimum of 4, is attained at the unique positive function u I ( x ,c). By (6.1.31), for c sufficiently small, say c < c,, 2 u , ( x , E ) are nondegenerate minima for (6.5.14). Consequently, for c < min(cO,1/Ai), the inequalities (6.5.10) imply that the boundary value problem (6.5.14) has another pair of critical points 2 u * ( x , E) which must necessarily change sign in 8 . To prove this last statement, assume (6.5.14) has only the solutions -+ u , ( x , c) and uo(x), for c sufficiently small. By virtue of (4.2.7) and (6.1.1) we may suppose t u I ( x ,c) are nondegenerate minima of the associated functional 4, ( u ) , and moreover for c2 # A;' and less than A;' we may assume that the (Morse) index of uo(x) is at least 2. Consequently, by the inequalities (6.5.10) M , > 2, and thus M , > Mo - 1 > 1. Consequently, either (6.5.14) possesses a nondegenerate critical point u ' ( x , c) of index 1 and thus distinct from uo(x) and t u , ( x , E ) or (6.5.14) has a degenerate critical point that must also be distinct from uo (x) or t u I ( x ,c). The oddness of u3 then implies that t u * ( x , E ) satisfies (6.5.14). Actually in Section 6.7 we shall greatly improve this result by showing for Ai < c - 2 < Xi+ (6.5.14) has i pairs of distinct nonzero solutions t u , ( x , c), t u,(x, E), . . . , t u i ( x , c). Of course to establish such a result it is important to find a general critical point theory that does not distinguish between degenerate and nondegenerate critical points.

++

,,

As another useful application of the relations of Morse (6.5.10) we apply them to sharpen our results (5.4.29) and (6.3.25) by giving estimates for the number of solutions of operator equations of the form (6.5.15)

Lu - Nu = f,

f

E

H = a Hilbert space,

where L is a self-adjoint Fredholm operator and Nu E C ' ( H , H ) is a uniformly bounded completely continuous gradient mapping with strictly convex antiderivative 92(u). Moreover, in accord with (5.4.29) we suppose ( N ( r a + x,), a ) < (+ ( a ) , a ) , where +(u) = % ;)

PN(ra),

a E {Ker L n llxll = l},

and

x , I K e r L.


365

Then we prove (6.5.16) Theorem Under the above assumptions, if f E Range(L - N ) , the functional 4,(u) = + ( L u ,u ) - % ( u ) - (f,u ) satisfies the compactness condition ( C ) and apart from a possible exceptional set of first Baire category the number of solutions of (6.5.15) is finite. Moreover, in this case, if the dimension of the linear subspace T on which the quadratic form (Lu, u ) < 0 i s j < co, then the following Morse inequalities hold: Mj

> 1,

Mj+l

-

Mj

( - l)'M,+;

=

-+ 1.

-1,

M,+* - M,,,

+ M, > I ,

dim T

2

i=O

Proof: To prove $lf(u) satisfies Condition (C) we shall prove (*) if + Nu, - fll--+O, then lIu,ll is uniformly bounded. Once this fact is established the complete continuity of N implies that, after possibly passing to a subsequence, Lu, converges strongly and u, converges weakly in U, so that u, + z2 strongly, and Condition ( C ) follows. Now we prove (*); we decompose H = Ker L G3 HI so that u, = z, + y n , z, E Ker L, y,, E HI. Then (1 Ly,/I is uniformly bounded since

f E Range ( L + N ) and 11 Lu,

IILy,(J< ( J L u ,+ Nu,

-fll

+ IINu, - f l l

G const.

by the uniform boundedness of N . Since L is a Fredholm operator, lly,ll is uniformly bounded. Next we show that ~ ~ (and z nconsequently ~ ~ Ilu,Jl) is uniformly bounded. To this end we use (5.4.29) to prove that since f E Range ( L N ) , (f,a ) < (@(a),a). Thus if we assume JJz,JI= Jlr,a,JJ -+ 00, while lly,lJ remains bounded, ( N ( z , +y,,), a)-+(cp(a), a) > (f,a). On the other hand, by assumption 11 Lu, + Nu,, - f l l +O, which implies ( N ( z , y,), a ) + ( f , a), so Jlz,l( stays uniformly bounded as required. Next, we again apply (6.3.25) and Section 3.1D to assert that Range ( L N ) is open in H and L N is a C ' nonlinear Fredholm operator of index 0, so apart from a possible exceptional set f of first Baire category the critical points of L N - f are nondegenerate and isolated. The complete continuity of N , the Fredholm property of L, and (*) then yield the finiteness of these solutions. Finally, in order to verify the Morse inequalities (6.5.10) we show that (4j'(u)x, x) is positive definite off the set T, where ( L x , x) 6 p11x1J2. Indeed for x E [TI'-,

+

+

+

+

+

( 4 ; ( u ) x , x) = ( L x , x) - ("(u)x,

2 P11x112- (

x)

P - e)11x1I2 = E l l x l l 2 .

Consequently the Morse indices of a critical point U of g j ( u ) must be less than dim T. On the other hand, g,(U + u ) + - 00 as I(uII + 00 on the linear

366


subspace S = { u I (Lu, u ) < 0 ) . This is plain for ( L u , u ) < 0 by the Fredholm property of L . Finally for u = R a E Ker L with llall = 1, (6.4.29) implies for f E Range(L - N ) that as R + 00, (f, a ) < ( N ( i i + Ra), a). Consequently lim !j,(E+Ra)= R-CC

lim R { ( f , a ) - ( N ( i i + R a ) , a ) } =

-00.

R-CC

Moreover, on the set T , the strict convexity of N ( u ) implies that ($;(ii)w, w ) < 0. Consequently, if we restrict g f ( u ) to [ T I L , we find $,(u)+ 00 as (lull + 00, so that the result (6.5.10) implies that the Morse inequalities are valid. From (5.1.9) (the reduction lemma), we conclude that the only contribution due to studying the critical points on T is to increase the Morse indices of a critical point by the integer j , and consequently the Morse inequalities as described in the theorem hold.

6.6 The Critical Point Theory of Ljusternlk and Schnirelmann 6.6A

Heuristics

It is often important to study the critical points of a given smooth functional independent of nondegeneracy considerations. Such a theory of critical points was formulated in the years 1925-1947 by the Russian mathematicians L. Ljusternik and L. Schnirelmann. This theory is based on determining a topological analogue for the minimax principles which characterize the eigenvalues of a self-adjoint compact operator L. Indeed, as mentioned in (1.3.42), if the positive eigenvalues of L are denoted A:, A,: . . . , arranged in decreasing order, and counted according to multiplicity, then (6.6.1)

A:

=

sup

min ( L x , x ) ,

[ S , _ , ] xEsfl-I

where S n P ldenotes the unit sphere in an arbitrary n-dimensional linear subspace Z of H , and [ S,- denotes the class of such spheres as 2 varies in H . Since the eigenvalues of L are precisely the critical values of the functional ( L x , x) on the unit sphere ax, = { x I llxll = l} of H , it is natural to extend (6.6.1) to general smooth functionals F ( x ) by finding “topological” analogues for the sets S, - I and [ S,- ,I. A fundamental result extending (6.6.1) to nonquadratic functionals can be found by supposing that 9R is a Hilbert manifold and n(A) is an integer-valued function defined on a class of closed subsets of 9TL with the properties: (i) n(A) = 1 if A is a point of %; n ( 0 ) = 0;

6.6 LJUSTERNIK-SCHNIRELMANN THEORY

(ii) (iii) (iv) (v)

367

n ( A ) 2 n(B) if A 2 B ; n(A u B ) < n(A)+ n(B); n ( A ) = n(A,), where A , is an isotopy of A ; there is a neighborhood U of A such that n( U ) = n ( A ) .

These properties are consistent since the trivial function n ( A ) = 1 if A # 0 and n ( A ) = 0 if A = 0 satisfies (i) - (v). Now we prove the following result (assuming, for the moment, the existence of an integer-valued function n ( A ) ) , which we shall use as a substitute for (6.6.1). 6.6B The minimax principle

In order to utilize the properties of n ( A ) to study the critical points of a functional F ( x ) defined on % and set %* = { x I F ( x ) 2 a , x E 9 2 ) for real a , we prove

'

(6.6.2) Theorem Suppose that c is an isolated critical value for the C real-valued functional F ( x ) defined on a Hilbert manifold % whose Frechet derivative is Lipschitz continuous, and that F ( x ) satisfies the compactness condition (6.5.2). Then for E > 0 sufficiently small, there is a neighborhood U of the set K, = ( x 1 x E % , F ( x ) = c, V F ( x ) = 0 ) and a deformation { l,}of %'+. - U, such that l,(W+'- U,) C a'-'. Proof: To define the deformation {,, we consider the solution x ( t , xo) of the system

(6.6.3)

dx

= - a(l1 V

F ( x ) l l )V F ( x ) ,

~ ( 0=) xg,

where a ( z ) is any C" function with a ( z ) = 1 for 0 < z < 1, a ( z ) = 2/z2 for z 2 2, and such that z * a ( z ) is monotone increasing for all z 2 0. By (3.1.27), the solution of (6.6.3) exists for all t since a(II V F(x)ll)ll V F(x)ll is uniformly bounded for x E % . Next, let U be a small neighborhood of the set K,. Clearly K, is compact by (6.5.2). We show that for some 6 > 0, U contains the set

Otherwise, there exists a sequence of points y, E %,yn fZ U , and numbers t, E [0, I] such that F(y,)-+c, V.F(x(t,,y,))+O as t,+ t.. Now by Condition (C), since IF(x(t,, ,y,))l is uniformly bounded, {x(f,,,y,)) has a convergent subsequence with limit y. (Clearly y is a critical point of F ( x ) on % .) After relabeling indices, we note that

y,

= x(-t,.x(t,,y,))~x(-t*,Y)=y,

i.e., y , converges to j , which contradicts the fact that y, fZ U .

368

6


+ a2

Finally, suppose that for 8 sufficiently small, F(xo) < c f and xo E N , a small spherical neighborhood of radius S about K,. We show that F ( x ( 1 , xo)) < c - 8'. Once this fact is established, setting r = S 2 / 2 and U, = N , n ' X C + ' ,we find the desired deformation ll(W+'- U,) C % - ' by setting {,(xo) = x ( t , xo) for xo E 'X . Now for xo E F - I [ c - S 2 / 2 , c S 2 / 2 ] and xo $E N,, we see that since a ( z ) z 2 is monotone decreasing and J j Vf(xo)J\2 c,

+

Thus F({,(x,))

< c + c 2 / 2 - c2 = c - r 2 / 2 .

(6.6.4) Minimax Theorem Suppose F ( x ) is a C2 functional defined on If F ( x ) satisfies condition (6.5.2) on a smooth Hilbert manifold and [ A ] ,= { A 1 A c %, n ( A ) 2 i} is nonvacuous, where n ( A ) satisfies conditions (i)-(v) above, then assuming the critical values of F ( x ) are isolated:

a.

(i) If finite (6.6.5)

ci =

inf sup F ( x ) [Ali x E A

is achieved and is a critical value of F ( x ) relative to %. (ii) If c, = c , + ~= . . = cItJ = c is finite, then n(K,) > j 1, where K, = ( x 1 x E H , F ( x ) = c, and x is a critical point of F ( x ) } . (iii) If some c, = 00, then sup, F ( x ) = 03, where K is the set of critical points of F ( x ) on %. (iv) (Ljusternik-Schnirelmann multiplicity theorem) Similarly, if inf, F ( x ) is attained and is also a critical value of F ( x ) finite, c, = supLAl, relative to %. If Z, = . = Z,+/ = Z is finite, n ( K F )> j 1. Furthermore, if some Z, = - 03, then inf, F ( x ) = - 00.

+

-

+

Proof: (i): If c, is finite and not a critical value, then condition (6.5.2) implies that for some E > 0, F - ' [ c z, c - E ] contains no critical points. Consequently by (6.6.2), we may deform W+f= { x 1 F ( x ) < c E ) into W-' = ( x 1 F ( x ) < c - E } . Hence every A c [ A ] , with A C W+'is deformed into a new set A' c W-' and by Section 6.6A(iv), n ( A ' ) > i so that A' E [ A ] , . Thus maxA.F ( x ) < c, - E , while c, = inf,,,, maxA F ( x ) < max,, F ( x ) = c, - r . This contradiction implies that c, is a critical value of F ( x ) . (ii): Next we suppose that c, = c , + ~= . . . = c,+, = c, and prove

+

(6.6.6)

n(K,) = n(%+')

-n(W-')

+

for some e

> 0.

First by (v), there is a neighborhood U IIK, with n(K,) = n ( U ) . Secondly

6.6

369

LJ LJSTERNIK-SCHNI RELMANN THEORY

by (6.6.2), there is a deformation {, of 9lt with {,(b9Rc" - U ) 5 W-'. Therefore by (6.6.2) and the properties of n ( A ) , n ( 9 R c - ' ) 2 n('Xc+' U ) . Now by (6.6.2) n(LW+')

u U ) Q n( - U) + n ( U ) < n ( 9 R - C ) + n(K,).

= n(

9L"'

9TLC+'

This establishes (6.6.6). Finally to prove (6.6.5), we observe that n( W+') since contains subsets A with n ( A ) > i j . On the other j 1. (iii): Suppose that supK F ( x ) = a < 00, where K is the set of critical is isotopic to EX, points of F ( x ) on $9.Then for some E > 0, since n(91La") = n ( % ) . Thus if i < n(4'm), c, = infiAl,supxEA,F ( x ) < a E < 0 0 ; which contradicts the fact that ci = 00. (iv): Repeat the proofs of (i)-(iii) but deform LX'-' into '9?2'+' for some E > 0.

'x+'

>i+j

+

+

+

As a simple application of formula (6.6.5) we prove:

(6.6.7) Theorem Suppose is a C 2 complete Hilbert manifold without boundary, F ( x ) is a C 2 real-valued functional defined on which is bounded from below and satisfies condition (6.5.2). Then F ( x ) possesses at least n( 9R ) critical points, where n( Em) is any integer-valued function defined on % and satisfying properties (i) n(??R) - .(a)=u(%). i=O

By our assumptions, Kc#consists of a finite number of points (x;,

. . . , xi)

370


for each i = 0, . . . , N , and by the above inequality, the total number of such points must be at least n ( % ) . Indeed, otherwise, by property (iii) of n( A )9

5

i= I

N

nW,)

M

x2

n(x,i)

i=l j = l

n ( A ) . Proof: Properties (i)-(iii) are immediate. To prove (iv), suppose that { , ( A ) is a deformation of A , and { , ( A ) C U where Bj is closed and contractible on %. If Ai = l;'(Bj), then A, is closed in A and hence in %. In addition, A is covered by U A i , while A ; is contractible in X ( i = 1, . . . , N ) since S; I A, is a deformation of A , into Bi and Bi is contractible. To prove (v), it suffices to prove that any closed set A contractible in X has a neighborhood U with cat, U = 1. Let 5, be a deformation of A to a point p , and let V be a neighborhood of p with contractible in %. By th_e homotopy extension theorem, {, can-be extended to a deformation {, of X . Then A = { ; ' ( p ) = {;(' V )= { - ' ( V ) . Let U be a neighborhood of A with 0 C f - ' ( V ) . Then cat,(u) < cat([,- '( < cat, = I since is contractible and closed in 92. Finally, we prove the maximality of cat, A . If cat, A = 1, A is deformable to a point p in 317,. Consequently, for any function n ( A ) satisfying properties (i)-(v), n ( A ) < n ( p ) = 1 = cat, A . If cat, A = N

?=,

v))

v

v

6.6

< 00,

A

=

U

LJUSTERNIK-SCHNIRELMANN THEORY

37 1

ys ,Ai,where each A j is deformable to a point pi E 9 2 . Thus

n(A,) = n ( p , ) = 1, so that

)

U A;
i}. Thus the following estimates, connecting the category of “nt to other properties of Lm, are particularly important. (6.6.9) Suppose A is a closed subset of a Hilbert manifold OX. Then:

+

1 (where dim A denotes the dimension of the (a) cat, A Q dim A set A ) ; (b) cat Gx > cup length ”X(see Appendix A for the definition of cup length); (c) if dim ?X = cup length u9R, cat “311. = dim 9R, 1; (d) let P k denote the k-dimensional real projective space, and P m ( X ) denote the infinite-dimensional projective space obtained by identifying the antipodal points of the unit sphere { x 1 JIxjJ= 1) of a uniformly convex Banach space X,then

+

cat,. P k = k

+1

for n

and cat,,(,,Pk(X)

=

k

+1

> k,

where

P k ( X )c P“(X).

Since these results are topological in nature, we omit their proof, 2nd refer the reader to Schwartz (1969). 6.6D

Application to nonlinear eigenvalue problems

The Ljusternik-Schnirelmann theory of critical points can be carried over with considerable success to manifolds ‘?It modeled on reflexive Banach spaces. A simple, yet nonetheless important, instance of such an extension is the study of nonlinear eigenvalue problems mentioned in Section (6.3A) in which case L3R, is a hypersurface. Let 1.3 (x) and 3 (x) be real-valued C 2 functionals defined on the reflexive Banach space A’,which we shall assume is uniformly convex. We wish to consider the nontrivial ’Another invariant n ( A ) satisfying ( i H v ) of Section 6.6A is the “genus” function introduced by Krasnoselski (1964). However due to the maximality property (6.6.8) we use cat A here.

372


normalized solutions ( x , A) of the equation (6.6.10)

&'(x) = A% '(x),

& ( x ) = const.

Clearly the solutions of (6.6.10) are contained in the set of critical points ( x ) on the level set = { x 1 x E X , &(x) = c of the functional 91 ( c = const.)}. If we suppose that & ( x ) is a Fredholm functional, then, by virtue of (3.1.47), apart from a set of real numbers of measure zero, &, is a Banach manifold modeled on the space X . We shall prove the following analogue of the spectral theorem (1.3.40) for linear self-adjoint compact operators defined on an infinite-dimensional real Hilbert space H .

aC

(6.6.1 I) Theorem The equation (6.6.10) has a countably infinite lA,,l .+m, and u, -+ 0 number of normalized solutions (u,,, A,,) with u,, E weakly as n -+00 for every real number c > 0, provided the operators @'(x) and 9 3 ' ( x ) satisfy the following conditions:

aC,

(i) & ' ( x ) is a C I odd gradient operator with & '(0) = 0, and for any x # 0, ( @ ' ( s x ) , x ) is a strictly increasing function of the positive real variable s; (ii) the functional @(x) is coercive, i.e., @(x)+ 00 as J J x--., J J00; (iii) whenever x,, -+ x weakly in X and ( @ x , , } converges strongly in X * , then @xn + @ x and x,, -+ x strongly in X ; and (iv) 93 ' ( x ) is a completely continuous gradient operator with 3 ' ( x ) = 0 if and only if x = 0. Proof: Consider the sets &c /Z2 obtained by identifying antipodal points of the level sets &, = { x I x E X , @ ( x ) = c}. Hypotheses (i)-(iii) show QC is nonvacuous; indeed since @ ( x ) =(;! & '(sx), x ) ds so that & ( k x ) is a continuous function of k with range [0, 00). The coerciveness of & ( x ) implies that eCis bounded in X , and consequently, since for x E gC, (@'(x), x ) > I l ( @ ' ( s x ) , x ) ds = c, Il@'(x)ll 2 0

C

=p>o.

~~PxEw,llxll

Hypothesis (i) implies that each set aCis starlike with respect to x = 0 since every ray through the origin, { t x 1 t E R', llxll = l}, intersects eCat precisely two points, ? t ( x ) x . Thus there is a one-to-one mapping of P m ( X ) = XZ,/Z2 (the unit sphere aX,of X with antipodal points identified) onto @.,/a2defined by setting f ( x ) = t ( x ) x . To prove that f ( x ) is continuous (and in fact differentiable), we note that for x E & c , c # 0,

d for s > 0. = (t?'(sx), x ) > 0 ds Thus the implicit function theorem implies that t ( x ) and, consequently,

- &(sx)

6.6

373

LJUSTERNIK-SCHNIRELMANN THEORY

f ( x ) are continuous. Therefore, W , / Z , is homeomorphic to P m ( X ) by

means of the mappingf. We now define a deformation of subsets of Qc/Z2 which has some of the properties of the deformation along gradient lines described in (6.6.4). For this purpose, note first that since X is uniformly convex, the duality' map J : X * + X is locally Lipschitz continuous. Then we consider the solution x ( t , xo) of the initial value problem dx = u + u ( x , u)J d x ,

x(0) = xo E

dt

W,,

where u is an element of X and a ( x , u ) E R' are to be chosen so that (i) &, and (ii) % ( x ( t , xo)) is a decreasing function of t . We determine a ( x , u ) by requiring that whenever x ( t , xo) satisfies the initial value problem then & ' ( x ( t , xo)) = c. Thus (W'(x),u + a ( x , u)J ax) = 0 and a ( x , u ) = (&'(x), .)/\I Q '(x)l12. We determine u by supposing that xo is not a critical point of 94 ( x ) restricted to W,. Then x ( t , xo) E

%(x(t,xo))-

:fi(~~)=I'(~,fi'(x(r)), c + u(x, u)J@"x)df

Setting V ? I ; ( x ( t ) )= ol; ' ( x ( t ) ) -

(:h ' ( x ( r ) ) ,J d ' x ) tP'(X) = J ' ( V ' q x ( l ) ) ,

I1 t-? '(x)I12

.) df,

0

where V % ( x ) denotes the gradient of $3(x) restricted to W,. Thus choosing u = - V $8( x ( t ) ) , we find that % ( x ( t , x0)) - %(xo) = -

I'

11 V

L!(x(r, xo))112dt.

We now proceed to prove that the numbers c,+ =

Inf

sup

91 (x),

c,- = sup inf 9(x)

AC[AIi x E A

[All A

are critical values of cT3 ( x ) / @ , , where [ A ] , = { A I A E &,/Z,, cat,JA) 2 i}. Clearly by (6.6.9), since & , / Z , x P m( X), the classes [ A ] , are nonvacuous and form a strictly decreasing sequence [ A ] ,IJ[ A ] , 3 [ A ] ,

'

The duality mapping J is defined to be the Frechet derivative of the functional I ( u ) = 4 ll~11~. For a uniformly convex Banach space, I ( u ) is differentiable on the complement of the origin, and J satisfies the properties ( J u , u ) = IIJUII

llu1l

and

IIJ(u)ll

=

llull

314


I . ...However, in order to repeat the argument in (6.6.4), we must verify the following analogue of Condition (C): (*) E

If c # 0, E

aC,and V (3x,

is sufficiently small, u, E 53 -'(c + E , c 0, then { x,,} has a convergent subsequence.

>O

4

-

€1, u,

To verify (*), we may suppose (after possibly passing to a subsequence) that (a) x,+Z weakly in X , (b) @'(x,) is weakly convergent, and (c)

11 d?'(x,J is uniformly bounded above zero and convergent. Then suppose

Since 3 ' is completely continuous, (c) implies that (33 '(x,), J B '(x,)) X @'(x,) is strongly convergent. Consequently, we may suppose that (93 '(x,), J @ ' ( x , ) ) is convergent to a real number p (say). The number /3 # 0 since otherwise (6.6.12) would imply that 93 '(x,)+O and by hypothesis (iv) that X = 0, which is impossible since by virtue of (iv), %-'[c + E, c - E ] is weakly closed and does not contain zero for E > 0 sufficiently small. Thus { @'(x,)} is strongly convergent, and by hypothesis (*), x, + x strongly in X . Consequently the desired result is established. Now we prove that each c,+ # 0 is a critical value of 53 ( x ) restricted to a CIndeed, . otherwise by (*), for some c > 0, 93-I[c,+ c, c,+ - e l contains no such critical points. Cons_equently, as in the proof of (6.6.2), we can find a deformation 5, and a set A c [ A ] , such that supA 3 (x) = c,+ + E , so tkat sups,(A, 53 (x) < c, - E. This is the desired contradiction, since cat ( { , ( A ) ) > i , implies that E [ A ] , and

+

.(,(A")

sup 53 ( x ) < c, - E . sdk) By reversing the procedure just given and deforming the sets [ A ] ,so that 93 ( x ( t , xo)) is increasing along x ( t , x,,), we may show that the numbers c,+ = inf sup % (x) [Ali

0. Then the equation (6.7.8) will have at least q distinct solution pairs 5 x,, n = 1, 2, . . . , q, provided the following two conditions are satisfied:

+

(a) F ( x ) = f Q(x) "X(x)is bounded below, and (b) F ( x ) > 0 for llxll sufficiently large. Proof: We consider the manifold Gslt obtained by deleting the origin from the Hilbert space H and identifying the antipodal points of H - ( 0 ) . In addition, '97-is a smooth manifold for each x bounded away from the origin. Clearly, 9R contains sets of Ljusternik-Schnirelmann category n = 1, 2, 3, . . . since the real n-dimensional projective space 'Yn( H ) c uX for each n. Now F ( x ) = 4 Q(x) %(x) is even in x and can thus be considered as a C 2 differentiable functional on % . Clearly, the functional F ( x ) satisfies Condition (C) on the stt Em-, = { x 1 F ( x ) < - E } for any c > 0 since if x E Gslt-, by hypothesis (b), IIxJ(is uniformly bounded so that if F'(x,)-+O for x, E Gslt-,, then the

+

6.7 APPLICATIONS OF THE GENERAL CRITICAL POINT THEORIES

379

sequence has a weakly convergent subsequence {x,} such that xq Lx, - N x , +0. Now the complete continuity of L and N imply that { xn,> is strongly convergent. Now we consider the numbers defined by e n ( % ) = infLvlnsup,, F ( x ) , where V is a subset of 9R such that cat( V , 3n)> n, and [ V ] , is the class of all such subsets. We show that for E > 0 and sufficiently small, all the numbers c0(9R), e l ( " ? ) , . . . , cV-,(%) are less than - c ; so that any critical points of F ( x ) lying on any of these levels is contained in "X-,. To obtain the desired bound, we observe that since the index of the quadratic form Q(x) = (x, x) - ( L x , x) is q, there is a q-dimensional subspace of H , Hv,and an absolute constant c < 0 such that for each x E Hv, Q(x)

cIlxl12.

Consequently, identifying the antipodal points of the sphere of radius R in Hv, we obtain a set q for each R > 0, so that T R E [ V ] , for each n = 1, 2, . . . , q. On the other hand, for small R and any X

€

qR,

(6.7.10)

F ( x )= Q(x)

+ %(x) < c l l x l j 2 +

0(1Ixl\~)< + c R 2 < 0.

Combining these two facts, we find that for R sufficiently small, inf sup F ( x) IVIn

v

< sup F(x) < 0. Here we derive some information on the number of solutions of (6.7.15) as A varies over the interval (0, m). Throughout this section we use the definitions and notations of Section 4.3B. We prove (see Fig. 6.3) (6.7.16) Theorem Suppose ( L u , u ) > 0 for u # 0, and the eigenvalues (with multiplicities of u = XLu are ordered so that 0 < A, < A, < A, < included). Then:

-

(i) for fixed A E (An-,, An], the equation (6.7.15) has a t least n - 1 distinct pairs of solutions (k u,), j = 1, 2, . . . , n - 1. (ii) Let R > 0 be a fixed positive number, and 9LR= { u I u E @*,,(a), 4 llul12 + 4 (Cu, u ) = R } . Then the equation (6.7.15) has a countably infinite number of distinct solutions ( u , ( R ) , A n ( R ) ) such that: (a) u , ( R ) E %, and A n ( R ) > A,; (b) A,(R)+m as n + m ; and (c) as R+O, (u,(R), A,(R))+(O, An). (iii) Suppose the multiplicity of An is k and A, < A,, then near (0, A,) there are at least k distinct one-parameter families of solutions i = 0, . . . , k - 1, such that ( ~ ~ + ~ ( eAn+i(e))+(O, ), An) as z +O, where E is proportional to 11 u , ( ~ ) 1 1 ~ .

-,

6.7 llxll

38 1

APPLICATIONS OF THE GENERAL CRITICAL POINT THEORIES

t Y

0

x2

AI

x3=x4

-

I

x

A5

x

FIG. 6.3 Hypothesized configuration for the origin of four solution pairs for h between h, and A,. (The “branches” emanating from h = hi and llxll = 0 denote nontrivial solution pairs for the equation (6.7.15))

Proof: The results (i)-(iii) are almost immediate consequences of the abstract theorems established earlier in this chapter. Proof of (i): The desired conclusion is established by applying Theorem (6.7.9) to the equation (6.7.15). In the present case, we replace the operator L in (6.7.9) by AL for fixed A E (Anpl, A,], and N in (6.7.9) by C. Thus F(u)= 1 1 ~ 1 1 ~A(Lu, u ) ) + (Cu, u). Clearly, the quadratic form referred to in (6.7.16) becomes Q ( u ) = (/u11’ - A(Lu, u); and since A E (A,- I , A,], the index of Q ( u ) is n - 1. Thus (i) will be established once we show that F ( x ) + co as ( ( x I I + 03 since then the provisions of Theorem (6.7.9) will be satisfied. This coerciveness property has already been proven in Section 6.2B. A simple proof for the present case follows from the estimate

+

:ll%(u, u)ll’ u ) , f ) + :I I % ( u , u)ll’.

F ( u ) = t llull’ - t A ( % ( u , f ) , u ) + =

;l/u11’-

+A(%(#,

+

Since 4 A(% ( u , u ) , f ) < c2 All % ( u , u)1I2 + (X/2~’)[lfll’ for any c > 0, we find that for AE’ = 4, F ( u ) 2 +llul12 - A’llfll’. Consequently, F(u)+co as J J u+ J Jco since A is fixed. Proof of (ii): The first part of the conclusion follows from Theorem (6.6.11). Indeed, setting @u = u Cu and % u = Lu, we note that the hypotheses of Theorem (6.6.1 1) are easily verified. For example, to show that f(s) = ( ~ ’ ( s u )u,) is a strictly increasing function of s E [0, 00) (for fixed u # 0), we observe that by virtue of the homogeneity of Cu,

+

f(s) = s(IuI12+ s3(Cu, u ) ,

382


so that ~ ( s= )

IIU/I’

+ 3 s 2 ( c u ,u ) > o

for u

z

0.

On the other hand, to verify hypothesis (iii) of (6.6.11), we see that if u, + u weakly and the sequence { &u,) converges strongly, then the complete continuity of Cu implies that u, + u strongly. The second part of the conclusion is obtained by using the fact that the solutions { u , ( R ) ) are characterized as critical points associated with the critical values c,( R ) = sup inf 4 ( L u , u), [Vln v where V is a symmetric subset of %, with cat(V, Gx, /Z2) > n and [ V ] , is the class of all subsets of Gx,. Thus since f lu11’ 6 R as R+O, llull +O; and consequently a slight variation of Lemmas (AHC) of (6.7.6) shows that as R+O, (u,(R), A,(R)+(O, A,). Proof of (Ill): The result can be proved on the basis of Theorem (6.7.6), by converting the equation (6.7.15) into an equation of the form (6.7.1), by scaling. Indeed, setting u = uv in (6.7.15), where u # 0 is a real number to be determined, we find that by virtue of the homogeneity of C, u satisfies the equation v + u’Cv = ALv. Setting u2 = A, we obtain u = A ( L v - Cu)

(6.7.17)

for A>0.

Hence applying Theorem (6.7.6) to (6.7.17), we obtain the desired families of solutions bifurcating from (0, A,). Comblned buckllng-bendlng problem As mentioned earlier in Section 6.2B, the equilibrium states in this case are identical with the solutions of the inhomogeneous nonlinear operator equation (for fixed A):

(6.7.18)

u

+ cu - XLU = f

(f # 0).

The considerations of (i) do not apply to this equation since the associated functional

(6.7.19)

9,,(u) =

f llu112 + j (CU,u ) - fX(Lu, u ) - (f, u )

is no longer symmetric with respect to the antipodal map, because f # 0. However, we now show that the Morse critical point theory of (6.5.10)is applicable to the functional J,,(u). First, we recapitulate some facts that will be needed in the ensuing argument.

(6.7.20) Lemma For fixed A, the operator A,,(u) = u + Cu - XLu is a Cm,proper nonlinear Fredholm mapping of index zero of k ,2(Q) into itself. Moreover, the singular values a,, of A,(u) form a closed subset that is nowhere dense in H . Proot: The properness of Ah(u) was established in (2.7.18), whereas the C m smoothness property follows immediately from the fact-that the operator Cu is homogeneous of degree 3. In addition, that a, is nowhere dense in W2,,(a) follows from Smale’s extension (3.1.45) of Sard’s theorem, once we show that Ah(u) is a nonlinear Fredholm operator of index zero. This fact follows immediately from (2.6.3)since A,(u) can be represented as a smooth compact perturbation of the identity.


383

We now state an application of our discussion in (6.5D) to the solutions of (6.7.18). (6.7.21) Theorem For each fixed h and almost allf E g2, 2(0)(i.e.,f E g2, 2(0)- OJ, the solutions of (6.7.18) will be a finite number of nondegenerate critical points of 4,,(u). These critical points will satisfy the Morse inequalities (6.5.10) by virtue of the properness of 4,’(u). More generally, for any fixed f, (6.7.18) is always solvable and the totality of solutions { w ) satisfies the a priori bound

(*)

IlwIl2,z

2(+~211Ffll12 + ;llf112>”2.

Furthermore, if 4,(u) has at least two isolated relative minima, then 4,(u) will possess a third critical point. By virtue of Lemma (6.7.20), the points f E ( g2, 2(0)- (PJ are everywhere dense in W 2 , 2 ( 0 ) and , A , i ’ ( f l must consist of nondegenerate critical points of 4,,(u). Thus the finiteness of this set A < I ( f l will follow from the properness of A,, and the implicit function theorem. Indeed, if there were an infinite number of points in A h ‘0,these points would lie in a compact set and so have a convergent subsequence, contradicting the fact that nondegenerate critical points are isolated. That these cfitical points of 4,(u) satisfy the Morse inequalities (6.5.10) when regarded as defined on H = W2,2(Q) then follows from the fact that the antiderivative of any proper gradient mapping defined on H automatically satisfies Condition (C) on H. The a priori bound (*) also follows from simple considerations. Indeed, if w satisfies 4,,(u) = 0, we find by virtue of (6.7.18), that

P!ool:

llwl12 + ( C ( W ) , w ) - WC(Ffl>w), w) = (f, w>.

Thus for any

c

> 0, the Cauchy-Schwarz llwl12 + I l C ( W ,

inequality implies

w)l12 - IhlcllC(w, w)Il2

-

Ihl T l l F O 1 l 2 4 llfll

IIWII.

Setting z = l/lAl, we find llw112 4 llfll I I W I I + I~1211~fl112~ from which (D) follows. The last statement in (6.7.20) follows from (6.5.3) and the fact that 4,,(u)+ co as llull+ co.

6.7D Stationary states for some nonlinear wave equations

We seek some time-periodic, complex-valued solutions of the special form u ( x , I ) = e’”u(x) for nonlinear wave equations of the form (6.7.22) or (6.7.23)

u,, = Au - m2u - iu, = A u

+f(x,

(ul’)u

+f ( x , 1 ~ 1 ’ ) ~

+

defined on R ’X R N . Here h is a real number, u ( x ) ( 0) is a real-valued, smooth function vanishing exponentially at 00, and f ( x , luI2) is a C ’ positive function of x that is odd in 1uI2. Such solutions are exact nonlinear analogues of stationary states for linear Schrodinger equations, and we therefore refer to these solutions as stationary states.

384

6


Substituting U ( X , t ) = e'"u(X) in either of the above wave equations, we find that U ( X ) satisfies the semilinear elliptic equations

(A - P ) u + f ( ~U')U , = 0,

(6.7.24)

x ERN;

where p = Xz - rn2 in the case of (6.7.22), and p = X in the case of (6.7.23). Thus to investigate the stationary states of either (6.7.22) or (6.7.23), we shall restrict the function f ( x , y ) and: (a) determine those values of p for which (6.7.24) has nontrivial solutions in L ~ ( R ~ ) , (b) for fixed p > 0 prove that (6.7.24) has a countably infinite number of distinct solutions in L2((WN). In answering both (a) and (b) we shall find special "nonlinear" phenomena. We begin by considering the question (a) in case either:

(I) for all u, O < f ( x , lul2)=g(x)(lul"}

with

O 0 and 0 < o < 4/(N - 2). This equation has a nontrivial solution w , Indeed, if we rewrite as an operator equation in H = @,,2(0, 00) with norm llwll; = JF(wz fib2) dr, the operator defined by the duality for

(t)

+

method as clj

grpnJwIou'@ with

(:%'(w),@)=/

O < a 0, there is a constant (Y < 4 with %xu) < a(%’‘(#),u ) for all u E W,,,(R’”).The reader will easily verify the remainder of the hypotheses of (6.6.1 1) for the operator 3’. If f ( x , u2) satisfies hypothesis (2), we proceed as in the proof of Theorem (6.7.25) and consider the equation

where w(r) = r ( ” ’ - 1 ) ’ 2 ~ (and r ) w(0) = 0. As before, the operator % ’ ( w ) defined by

is a completely continuous gradient mapping of l@l,2(0,co) into itself. Moreover, as in the above paragraph, the hypotheses necessary to apply Theorem (6.6.1I ) are then easily verified. Thus the theorem is established after rescaling the associated eigenvalue as in (ii) above. 6.7E Geodesics between two points of a compact Riemannian manifold

Let (972“g) be a compact, smooth, Riemannian manifold of dimension N with metric tensor g. Then, in terms of local coordinates, the geodesics between two points a and b of ‘XNare the solutions (passing through a, b ) of the following second-order system of ordinary differential equations (6.7.29)

d2xk fit2

N

+

C r;(x)

dx’ dxJ dt dt = 0,

( k = 1,.

. . ,N ) .

‘,j=1

Alternately, these geodesics are the critical points of the arc length funcds, relative to all the smooth curves of 92’’’ passing through a, 6. tional, This latter characterization of geodesics and its variants are useful in studying the structure of the geodesics joining a and b. The simplest basic result is due to Hilbert.

1:

(6.7.30) There is a geodesic of minimal length joining the two points a and 6 , on a smooth, compact, (connected) Riemannian manifold.

(a”’,

Proof: We consider the class of rectifiable curves KO, on g) joining a and b. Since 3nN is connected, this class is nonvacuous. We


387

parametrize the curves ( c ) of K,, by the parameter r = s/ L , where L is the length of a curve in Ka,b and s is the arc length measured from the initial point a. With this parametrization, the desired geodesics are critical points of the functional 4 (c) =jh ds2. Clearly, by the properties of the Lebesgue integral, 9 (c) is lower semicontinuous with respect to uniform convergence in KO,b . Let 4, = inf g (c) over KO,b . We will show that 4, is attained by an element co of KO,b.To this end, it suffices to show that the set of curves in KO, with bounded length, is compact. Thus let { c,,(T)} be a sequence of curves in Ka,b with length L(c,) < M (say). Then for any T ~ T~ , E [0, I] and fixed n, (6.7.31)

d(C,(71)? cn(.2))

= L(C,)b,

- 721

< MI71- 7-21,

where d ( x , y ) denotes the Riemannian distance between x and y . Thus {c,,), regarded as continuous mappings from [0, I]+(%’”, g), are uniformly bounded and equicontinuous. By the Arzela-Ascoli Theorem, { c,,) has a uniformly convergent subsequence whose limit co will again be a rectifiable curve, since, by (6.7.31), d(cO(T1), ~ ~ ( 7< ~ ) M)) T -~ T ~ ) . A deeper study of the geodesics joining a and b can be carried out by utilizing the results of 6.5 and 6.6. Indeed, suppose ( LXN, g) is isometrically embedded as a closed submanifold into a Euclidean space lRk(’”) of sufficiently high dimension. Let Wl, 2([0, I], ( 8)) be the closed subset of the Hilbert space N = W,,,([O,I], R k ( N ) )consisting of those elements C ( T ) E H for which the image c[O, 11 c L9RN.Wl,2([0, 11, (EmN, g)) is a Hilbert manifold, as is its closed subspace a(ERN ; a, b ) consisting of those g ) ) satisfying c(0) = a and c(1) = 6 , elements C ( T ) of Wl,2([0, 11, (”XN, with a , b fixed points of ERN. Now let ( c ( T ) , ~ ( 7 ) )denote the length of the vector ~ ( 7 ) .Then the critical points of the functional Y(c) = J ; ( ~ ( T ) c(7)) , d7 = J c ds2 in the space a(”XN ; a , 6 ) coincide with the g ) joining a and 6 . If EmN is homeomorphic to the geodesics on sphere S N , the result (6.5.12) can be used to show that there are an infinite number of distinct geodesics joining a and b on (%’”,g). Indeed one can show that the functional g(c) satisfies the condition C, (6.5.2), on a( 9RN ; a , b). Then the result follows provided it can be shown that the Betti numbers of s 2 ( 9 K N ; a , b ) are different from zero for an infinite number of distinct integers. Indeed, it is known that the Betti numbers R j of s2( 9RN; a , b ) form a periodic sequence of length ( N - 1) consisting of the numbers 1 when i = 0 (mod ( N - 1)) and 0 otherwise. Note B at the end of this chapter mentions other interesting information on this topic. Since an adequate treatment of this topic is contained in other monographs, we refer the reader to the books of Morse (1934) and Schwartz (1969) and the article of Palais (1963) for further information.

a’”,

(a’”,

388


NOTES A The Dlrlchlet problem for parametrlc surfaces of constant mean curvature Let D be a simply connected domain in the u-u plane with boundary a0. We wish to determine a parametric surface S = ( X ( u , u) = x,(u, u), xz(u, u), x3(u. u)) of constant mean curvature M defined over 0 , which assumes prescribed smooth boundary values f on a 0. The system of partial differential equations defining S are

Here aX/au r\ax/a; denotes the vector product of (ax,/au, axZ/au, ax,/au) and (axl/i3u, ax,/au, ax,/au). The problem can be simplified by setting X ( u , u) = F(u,u) Y ( u , u), where F(u, u) is a harmonic 3-vector in 0 satisfying the boundary condition (a). Then we seek a C 2 3-vector Y ( u , u) defined over 0 that satisfies (*) and the homogeneous boundary condition Y I a n = 0. To determine the function Y , we suppose that H < $ R and minimize the functional

+

J( Y )=

/k( I

V Y l2 + M Y .

(

A

) ) du do

over the set W , = Fkl,,(0)n Z, where g R denotes the continuou: vector functions Y on with ess supn IY(u, o)l G R . Clearly, if Y E Wl,,(0) n Z, and I YI,_ < R attains the infimum of 9 ( Y) over W,, then for all test functions 3 E Wl, 2(0)n L,(P),

Consequently, by the regularity results mentioned in Section 1.5 Y will satisfy (*) as well as the homogenecus bounsary condition Y I , , = 0, provided we show that the desired infimum is attained at Y with 1 YI < R . Actually we prove Theorem Let IMI < 1 and suppose that sup,, (**) as well as the bound supn 1 x1 < 1.

If1

G I . Then (*) has a solution satisfying

The result is obtained in twp steps by first showing that the minimjzation problem mentioned above has a solution Y ( x ) , and secondly by proving a bound for Y ( x ) for IMI < 1. For the proof we refer the reader to the paper of Hildebrandt and Widman (1971). B Marston Morse’s results on geoderlcs between two polnts P , Q on a compact Rlemannlan manlfold ( M , g) Thegeodesics on S N with its metric of constant curvature are easily found for two points

(P,Q) of S N that are not antipodal; these geodesics are in one-to-one correspondence with the integers, and when ordered by length and denoted go, gl, . . . , g, . . . , the Morse index of g, is easily found to be (N - 1)n; here we can associate the integer n with the number of times g, contains the antipode of P in its interior. This result has the following generalization for the geodesics between the two fixed points P , Q of (Gm, g) with dim Gm = N : (i) The arc length functional J has critical points if and only if the critical point corresponds to a geodesic of (%, g) parametrized by arc length; and the geodesics between P

NOTES

389

and Q are all nondegenerate if and only if Q is not a conjugate point of P, and moreover form a set of measure zero of points of (nt,g). (ii) Provided the points P and Q are not conjugate relative to (%, g), the critical points of J all have finite Morse index, and this index is exactly the number of conjugate points (counted with multiplicity) of one endpoint in the interior of the associated geodesic. C Use of the calculus of varlatlons in the large for calculations of homotopy groups

The results of Section 6.5 on infinite-dimensional Morse theory were used in Section 6.7 to obtain results about the existence of critical points from topological information. Actually the converse procedure of utilizing knowledge of the critical points of a standard functional (say the arc length functional) defined on a compact manifold Gnz. to determine facts about the topology of % has proved remarkably successful. Many interesting results concerning homotopy groups can be achieved in this way. Thus by utilizing the results of Note B the first part of the Freudenthal suspension theorem (1.6.8) concerning the homotopy groups of spheres can be proven. Much information on the homotopy theory of the classical Lie groups was obtained in this way by Bott and Samelson. For the details of these results we refer the interested reader to the monograph of Milnor (1963). A typical result in this connection is the following periodicity theorem for the homology of the loop space S2(SN):

H , ( w s N ) )= H

0

q=O

(modN-I),

otherwise.

This periodicity phenomenon is also true for the homotopy groups of many Lie groups as was discovered by Bott. Moreover, from (*) one can compute the cohomology ring H.(S2(SN)) mentioned in Appendix A and conclude from the general critical point theories of Sections 6.5 and 6.6 that for every Riemannian metric on S N and two distinct points P, Q E S N , there are an infinite number of geodesics joining P and Q.

D Applications of Ljusternik-Schnlrelmann type lnvarlants to equlvarlant mappings

In Section 6.6 topological invariants (such as the category of a set) were used to investigate the existence of critical points of an even functional. Such topological invariants can also be used to study mapping properties of other equivariant operators. An example is the following generalization of the Borsuk-Ulam theorem [Holm and Spanier, 19711. Theorem Let C be any compact mapping defined on the unit sphere a Z = ( x I x E X, llxll = 1) of a Banach space X,such that for f = I + Cf(aZ) lies in a subspace of codimension k of X, then dim(x I f ( x ) = f ( - x ) , x E a X ) > k - I . E Blbllographlc notes

Secfion 6.1: The results contained here are generalizations of the so-called direct method of the calculus of variations. These methods can be traced back to Lebesgue (1907) and Hilbert (1900). The notion of lower semicontinuity in numerous forms was investigated in the works of Tonelli (1921, 1923). The compactness Condition (C) stated in the text was first stated in Palais and Smale (1964). The illustration of the minimization methods contained in Theorem 6.1.8 can be found in Berger and Schecter (1977), while (6.1.20) can be found in the

390


book Ladyhenskaya and Uralsteva (1968). The results in equation (6.1.30) are contained in the paper Berger and Fraenkel (1970). Section 6.2: The result (6.2.5) is adapted from Berger (1975) where a more complete discussion of its relation to algebraic manifolds is given. Our discussion of flexible plates and shells can be found in Berger (1967, 1971). Plateau’s problem is well discussed in Nitsche (1974). Our proof is a modification of Garabedian (1964). The result (6.2.28) is contained in an unpublished paper of Berger and Wightman. Section 6.3: The results of this section can be found in Berger (1973) and Berger and Schechter (1977). Section 6.4: Our discussion of large amplitude periodic solutions of Hamiltonian systems is based on Berger (1971b). and its application to the perturbed Kepler problem is based on unpublished work of Berger and Arensdorf. Our discussion of metrics of prescribed curvature is based on Berger (1975) and Yamabe (1960). These results have been extended by Kazdan and Warner (1975) and by Moser (1973). Our discussion of steady vortex rings can be found in Fraenkel and Berger (1974).

Sections 6.5 and 6.6: Our discussion of Morse theory in Hilbert space is adapted from Rothe (1973) and Smale (1964). A more detailed reference is the paper Palais (1963). The illustration (6.5.16) can be found in Berger and Podolak (1977). Our discussion of the critical point theory of Ljusternik and Schnirelmann can be found in papers by Schwartz (1964) and Palais (1966). and in the book of Ljusternik (1966). A reference for the application of this result to nonlinear eigenvalue problems is Browder (1965). Proof of (6.6.1 I) is due to Amann (1972). Section 6.7: The application of the Ljusternik-Schnirelmann theory to bifurcation theory is due to Berger (1970). Later papers on the subject are due to Bohme (1973) and Riddell (1975). The idea contained in (6.7.9) is due to Clark (1973). For the result (6.7.11) see Ambrosetti (1973). The application of general critical point theories to nonlinear elasticity can be found in the papers of Berger (1974). Our discussion of nonlinear stationary states is based on Berger (1 972).

F On the multlpliclty preservation theorem (6.7.6)

This theorem reiterates the importance of utilizing global topological methods in studying degenerate critical points as contrasted with the algebraic methods of Thom’s catastrophe theory. The result has particular importance for the study of “nonlinear” normal modes of a Hamiltonian system near a singular point as described in Chapter 4. In that case, the violation of the Liapunov irrationality conditions associated with jth normal mode (say) of the linearized system implies the eigenvalue Aj associated with this mode is not simple. The number of such violations can be interpreted as the multiplicity of Xi, and the preservation theorem implies that the jth normal mode is not destroyed by the nonlinear Hamiltonian perturbation but merely distorted with the preservation of periodicity. This approach has been described in the papers Berger (1969, 1970a). Alternate finite-dimensional approaches to this problem have been carried out by Weinstein and Moser. As mentioned in the text, the virtue of the multiplicity preservation theorem is that it provides a method of “continuing” these nonlinear normal modes to large amplitude. Further results await research on the continuity of the eigenvalue “branches.” It can be shown, for example, that the period of the solutions described in Theorem (6.4.2) tends with decreasing amplitude to the smallest nonzero period of a linearized system (if nontrivial).

APPENDIX A

ON DIFFERENTIABLE MANIFOLDS

A set G3n is a manifold of dimension N if u31t is a Hausdorff topological space a n d each point x E 9R has a neighborhood W, that is homeomorphic to an open subset of RN.The set 9R is called a differentiable manifold (of class C k )if .%can be covered by a family of open sets 0, (called coordinate patches) each homeomorphic (by a C" map h,) to an open set in RN and such that on the intersection of any two such coordinate patches the change of coordinate mappings h h - I : hp(O, n " P O p ) - + R Nis a smooth map of class C k . A differentiable manifold 9RN is called orientable if it can be covered by coordinate patches 0, as above such that the change of coordinate mappings h a h i ' have positive Jacobian. Connected one-dimensional differentiable manifolds can be easily described. Any connected differentiable manifold of dimension 1 is diffeomorphic to a circle or some open interval of real numbers. A subset V of a differentiable manifold 3nN is an r-dimensional submanifold of EmN if there is a family of coordinate patches ( O , } covering such that (0, n V ) is a system of coordinate patches covering V and if x = (x,, x2, . . . , x N ) are local coordinates in O,, then

aN

0, n V = { x I xR+

, = x , , ~ = . . . = xN = 0}.

For the solution of problems in differential geometry, it is essential to study calculus on manifolds. For some considerations, this can generally be achieved by introducing local coordinates (xI,. . . , x N ) (regarded as a point of R N ) in a coordinate patch 0, of M N . Indeed, for x E 0, each map h,: 0, + RN can be written h , ( x ) = ({,(x), 12(x), . . . , {,(x)) = ( X I , x2,

. . . ,XN).

Moreover, x = ha-'(x,, . . . , x N ) . It is also useful to introduce the notion of a partition of unity on M N (i.e., a locally finite covering 'V of M and a collection of real-valued nonnegative smooth functions fv, V E Y on M such that the support of fv is contained in V and CVEVfv (x) = 1). Thus 39 1

392

APPENDIX A

to define the integral of a real-valued function g on M N with respect to a volume element d V , let (?,A) be any partition of unity on M N , and set

Now each term on the right can be evaluated by using local coordinates since the 7 can be chosen subordinate to the coordinate patches 0,. Moreover, this definition is independent of the partition of unity used since if ( W,, +,) is any other partition of unity (subordinate to O,),

The use of differential forms is essential for the solution of many problems in differential geometry (cf. Section 1.2A). Local coordinates are useful in defining differential forms on a manifold M N . Indeed, one can begin by defining differential forms on an open set D c [WN and then extending the definition to M N . We define differential p-forms of class C k on an open set D of R N (denoted /$)(&I)) to be the formal expressions w=

2

c i l j 2 ...$(x) dx;, A dxj2A

*

*

A dxiN

i, 0 such that c(w, w )

< (dw, d o ) + ( d T w ,dTw).

We now define Green’s operator G(w) for any w E A\,”(%) as the solution u of AU = - H ( w ) . The solution of this equation is unique, and in the Hilbert space 3Cp (%) obtained by completing A,( Em) relative to the norm llw112 =

( d o , dw) + ( d T q dTw),

the solution u exists (in the generalized sense), and by the ellipticity of A, in the classical sense. Thus we find for w E &(a), the Hodge-Kodaira decomposition

+ dTdC(w)+ H ( w ) .

w = ddTG(o)

On a complex manifold Em one may introduce a real analytic Riemannian metric g that in terms of the distinguished complex analytic coordinates can be written as ds2 = 2 2 G~dzj dFk. j,k

Such a metric is called Hermitian. A Hermitian metric may be introduced on any compact complex manifold. A Hermitian metric on %

gjk dzj A dZk

g =2 j,k

is called Kahler, if dg = 0. In that case, the complex Laplacean 0 is written

of=zgf a ’flazi 83, and the following result holds

If (%,g ) is a Kahler manifold, 0 defined as above satisfies n= A, where A is the Laplace-Beltrami operator defined on the real 2 N dimensional Riemannian manifold (%, g ) . In the general Hermitian case, we may again define the Hodge-Kodaira decomposition by following the argument used in the Riemannian case, with 0 replacing A. Indeed, if the (0, q)-forms are denoted &,, q ( % ) , and the harmonic (scalar or vector valued) (0, q)-forms are those (0, q)-forms w for which O w = 0, then we have a Green’s function G such that

--

+aTac(o) + H ( w ) , where H ( o ) is the harmonic part of w and G(o) is the unique solution u of n u = w - H(o). 0 = aaTG(o)

398

APPENDIX B

We note an L, scalar product ( , ) for forms of the same type and correspondingly, a formal adjoint $ T of 3 defined by ( 3 T w . w , ) = (0, $0,).

g T maps /\p,q(Gs17,)+/\p,q-,(Lm)

and ($q2= 0. In addition, the complex

Laplacean 0 =a g ' + j q preserves /\p,q(Gsn), commutes with 3 and gT, and is a strongly elliptic partial differential operator on forms of any fixed type. For a discussion and proof of these results we refer the reader to the book by Kodaira and Morrow (1971).

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INDEX

A Airy stress function, 11 Analysis classical linear, 215 nonlinear functional, 215 Analytic implicit function theorem, 134-136 Analytic operators, 84-88 majorant method of, 133-137 Anharmonic oscillator, perturbation and, 205 Approximate inverse, 137 Approximation asymptotic, 193 successive, 60, 111-127 Approximation problem, 216 A priori bound principle, 23, 210 Arzela-Ascoli theorem, 32, 92 Asymptotic approximation, 193 Asymptotic expansions, 193-204 validity of, 194-199 Asymptotic linearity, 230 Atiyah-Singer index theorem, 108 Atlas of class C‘, 356 Autonomous systems, continuation of periodic motions of, 291-293 B Banach manifold of class C ’ , 357 Banach-Mazur proof, 221 Banach-Mazur theorem, 296 Banach scale, 26 Banach space(s), 25-26, 88-89, 111 analytic implicit function thcorem and, 134 approximate inverse and, 137

asymptotic expansions and, 198 in bifurcation theory, 154 compactness and, 31 compactly homotopic elements in, 244 complex analytic element in, 252 of continuously differentiable functions, 26-27 Dirichiet problem and, 202 finite-dimensional, 121 Fredholm mapping and, 125 Galerkin approximations in, 231 of Holder continuous functions, 27 infima and, 300 isoperimetric problems and, 123 linearization in, 2 17 local surjectivity and, 119 odd finitedimensional, 281 one-parameter family of compact operations defined on, 270 Rayleigh-Ritz approximation and, 240 reduction lemma and, 226 Banach space analogues, of inverse mapping and implicit function theorems, 113 Banach theorem, 34 Betti number, 363 Bifurcated branches, linear stability and, 21 1 Bifurcation analytic set in, 192 of complex structures on compact complex manifolds, 188-192 for general operator equations at odd multiplicity eigenvalues, 21 1 points in, 15 1 subcritical or supercritical, 162-163 Bifurcation equations, 155, 162 symmetry assumptions and, 21 1-212 Bifurcation phenomenon, 14, 19 “buckled states” in, 178

409

410

INDEX

buckling phenomena in nonlinear elasticity and, 177-182 secondary stationary flow and, 183-186 specific, 173-192 Bifurcation theory, 9 , 6 0 applications of, 191-192 Brouwer degree in, 164-167 constructive approach to, 149-163 critical point theory and, 375-378 definitions and basic problems in, 150-151 heuristics in, 163-164 higher multiplicity and, 161-163 LjusternikSchnirelmann theory in, 375378 Morse type numbers in, 171-172 simple multiplicity in, 155-158 subcritical or supercritical bifurcation in, 162-163 transcendental methods in, 163-172, 213 Biharmonic operator, 11 Bilinear forms, self-adjoint operators and, 37 Boundary conditions “coercive,” 295 natural, 17 Boundary layer phenomena for Dirichlet problem (He), 200-204 for semilinear Dirichlet problems, 212 Boundary value problems coercive semilinear elliptic, 293 critical points in, 364 nonlinear free, 287 nonlinear operators and, 283-295 Bounded linear functionals, 30-31 Bounded linear operators, 32-35 special classes of, 35-39 Boundedness, continuity and, 64-65 Brouwer degree, in bifurcation theory, 164-167 Brouwer futed point theorem, 54 Buckling phenomena, thin elastic shells and plates, 177-1 80 L

Calculus elementary, 64-76 higher derivatives in, 72 mean value theorem in, 70 multilinear operators in, 70-72 Calculus of variations, 6 0 inequalities and, 42 nonlinear differential systems from, 16-18 Calderon-Zygmund singular integral operator, 33

Carathhodory continuity conditions, 76 Cauchy-Kowalewski theorem, 148 Cauchy majorants, 133 Cauchy-Schwarz inequality, 318 Center problem, 146 Chain rule, in elementary calculus, 69 Clarkson’s inequalities, 28 Classical solutions, of differential systems, 47-48 Closed range theorem, 34 Coercive boundary conditions, 295 Coerciveness, criteria for, 302-303 Coerciveness restriction, removal of, 236239 Coercive semilinear elliptic boundary problems, necessary and sufficient conditions for solvability of, 293-295 Cohomology groups, 57-58, 393-394 Compact differential operators, 92-93 Compact homotopy, 249 Compact linear operators, 35 Compact mappings, extension theorem for, 91 Compactness, 31-32,303-304 Compactness theorem, 42-43 Compact operators, Schauder fixed point theorem and, 90 Compact perturbations of continuous mapping, 244-246 of identity mapping, 246-256 of linear Fredholm mapping, 256-262 oneparameter family of, 272 Compact Riemannian manifold, geodesics between two points of, 386-388 Complex analytic Fredholm operators, 136-137 Complex Laplacian, 3 14 Complex manifolds, 394-395 bifurcation of complex structures in, 188-192 Complex structures deformation of, 8-9 nontrivial deformations of, 188 Composition operators, 76-77, 88-93 Cone preserving mappings, 298 Cone preserving operators, properties of, 281-283 Constant mean curvature, hypersurfaces of, 297 Continuation problems in bifurcation theory, 152 for nonlinear operators, 275-276 for periodic solutions of Hamiltonian systems, 298

41 1

INDEX

Continuity boundedness and, 64-65 Carathkodory, 76 complete, 65 Continuous mapping, compact perturbations Of, 243-246 Contraction mapping, 111-113, 117 Contraction mapping theorem, 11 2 Convergence, weak, 31 Convergent iteration scheme, 158-160 Convolution operators, 142 Couette flow, 186 Covering homotopy property, 218, see also Homo topy Covering space, of consistent Hausdorf topological space, 218 Critical point(s) degenerate and nondegenerate, 35 5-35 8 index of, 355 Critical point theory, 58, 167-171 application of, 375-387 for gradient mappings, 299-390 of Ljusternik and Schnirelmann, 366-374 O f M. Morse, 58,353-366 minimization problems in, 299-312 Cup length, 394 Cup product, 57

D Decay, amplification of, 385 Decomposition, Hodge-Kodaira, 396-398 Deformable plates, 318 Deformable shallow shells, 318 Deformation, of complex structures, 8 Deformation problem, 216 Demicontinuity, 65 De Rham cohomology groups, 393-394 Diffeomorphism, 7 Differentiable manifolds, 39 1-395 Differential equations, Euler-Lagrange, 1617 Differential forms, Hodge-Kodaira decomposition for, 396-398 Differential-geometric problems, 3-9 Differential operators, 77-79 duality and, 82-84 as proper mappings, 105-106 representations of, 80-84 Schauder inversion and, 81-82 Differential p-forms, 392 Differential systems classical and generalized solutions of, 47-51

linear, see Linear differential systems nonlinear, see Nonlinear differential systems Differentiation, in elementary calculus, 67-70 Dimensional analysis, 41 integral inequalities and, 61 Dkichlet problems boundary layer phenomena for, 212 positive solutions for, 285-286 for quasilinear elliptic equations, 283-285 for quasilinear elliptic partial differential equations, 297 semilinear, 49 solvability of, 284 Dkichlet problem &), application of, 200-204 Distribution solutions, 47 Domain theorem, invariance of, 269 Duality mapping, as FrBchet derivative, 372 Duality method, 82-84, 108 Dynamic instability, 15 in Euclidean quantum field theory in mean field approximation, 322-324

E Economics, mathematical, 60 Elastica problem, 11, 177 Elasticity, nonlinear differential systems and, 11 Elastic plate, flexible, 380-383 Elliptic boundary value problems, coercive, 293-295, see also Boundary value problems Elliptic equations, quasilinear, see Quasilinear elliptic equations Equation of motion, Euler’s, 286 Equilibrium, pear-shaped forms of, 151 Equivariant mappings, LjusternikSchnirelmann invariants in, 389 Essential properties, in compact homotopy, 245 Euclidean field theory, 14 Euclidean quantum ficld, dynamic instability in, 322-324 Euler ehsticu problem, 11, 177 Euler equations of motion, 12, 286 Euler-Lagrange differential equation, 16-1 7, 310, 320 Euler-Poincari characteristic, vanishmg, 340 Existence problem, in bifurcation theory, 152 Expansions, asymptotic, see Asymptotic expansions

412

INDEX

F

finite-dimensional approximations for, 231-243 Fibration, 296 fiied point theorems for, 269-272 global homeomorphisms for, 218-225 Field theory, 14 global theories for, 217-298 Finitedimensional problem, reduction of homotopy in, 243-246, 266-283 bifurcation theory to, 154-155 linearization of, 217-231 Finitedimensional spaces, mapping benecessary and sufficient conditions for tween, 51-59 solvability in, 277-281 Fixed point theories, 60 nonlinear boundary value problems and, for general nonlinear operators, 269-272 283-295 Flexible elastic plate, global equilibrium states Rayleigh-Ritz approximations in, 239-24 1 and, 380-383 spectral properties of, 272-276 Fluid stable homotopy class and, 279 ideal incompressible, 1 2 sujectivity properties for, 266-267 viscous incompressible, 1 3 Frechet derivative, 67-68, 123,217 univalence and homeomorphism properties duality mapping as, 372 Of, 268-269 Frhchet differentiable operator, 67, 72 General smooth mappings, steepest descent Fredholm mapping, 125, 265 method for, 132 linear, 256-262 Geodesics Fredholm operators, 36, 226 on manifolds, 3 complex analytic, 136-1 37 between two points of compact Riemanndifferential, 101 ian manifold, 386-388 linearization of, 296 Geometric measure theory, 6 2 nonlinear, 99-101,126-127,147-148 Geometry and physics, isoperimetric probproper, 263-265,296,298 lemsin, 337-353 self-adjoint, 277, 353, 364 Gibbs internal energy function, 16 Fredholm orthogonality condition, 333 Global equilibrium states, of flexible elastic Free boundary problem plate, 380-383 nonlinear, 12,13, 287 Global free boundary problem, 348-353 reformulations of, 349-350 Global homeomorphisms, 218-225 Freundenthal suspension operator, 256 Gradient mappings, 93 Froude number, 288 bifurcation theory for, 375-378 critical point theory for, 299-390 G multiple solutions of operator equations involving, 378-380 nonlinear eigenvalue problems for, 325-331 Galerkin approximations, for general nonspecific, 97-99 linear operations, 231-234, 241 steepest descent method for, 127-133 Garding’s inequality, 46 Gradient operators, 94-99 Gateaux derivative, 68 Rayleigh-Ritz approximations for, 239-241 higher-order, 72 semilinear, 331-336 Gateaux differentiable operator, 68, 86 Gravitation, relativistic theory of, 15-16 Gauss-Bonnet theorem, 224, 341, 345 Green’s function, 397 Gaussian curvature conformal metricg of, 345 H as constant, 6 Generalized inverse function theorems, -137-145 Hadamard proof, 222 Generalized operator theory, 216 Hadamard theorem, 224,226,313 General nonlinear operators Hahn-Banach theorem, 30 coerciveness restriction in, 236-239 Hamiltonian perturbation, 24,62 complex analytic mappings of, 274-275 Hamiltonian system continuation problems for, 275-276 nonlinear, 23 continuation of solutions in, 283 periodic motions of, 17-18

41 3

INDEX

Hartog’s theorem, 85-86 Hausdorf topological space, 218 Helmholtz singular vortex, 1 3 Hermitian connection, 314 Hermitian metric in complex manifold, 397 of constant negative scalar curvature, 313-317 Hermitian scalar curvature, 8 Higher multiplicity, in bifurcation theory, 161-163 Hilbert manifolds, 363, 369 Hilbert space, 25-26, 37, 271, 294, 300, 350 critical point theory in, 353-366 Fredholm operator mapping of, 277 infinite dimensional, 372 Morse inequalities and, 361 Morse’s lemma and, 355 real, 184-185 Hill spherical vortex, 1 3 HiIl vortex ring, 348 Hodge-Kadaira decomposition theorem, 190 for differential forms, 396-398 Holder continuous functions, 27 Holder inequality, 28 Holder inverse inequality, 291 Holder spaces, pointwise estimates in, 40 Holomorphic functions, mapping properties of, 8 Homeomorphism, for general nonlinear operators, 268-269 Homology groups, singular, 56-57 Homotopy, 243-246 compact, 244-245 for nonlinear operators, 266-283 stable, 246, 25 6-26 2 unstable, 261 Homotopy groups, 54-56 calculations of, 389 unstable, 56 Hooke’s law, 11 Hypersurfaces, of constant mean curvature, 297 I Identity mapping, compact perturbations of, 246-256 Imbedding operators, 34-35 Implicit function theorem, 115 Incompressible fluid, ideal, 12 Index of critical point, 355 Morse, 171, 378

in nonlinear Fredholm operator, 99 Index theorem, 108 Inequalities and estimates, in nonlinear systems, 39-47 Inessential properties, in compact homotopy, 245 Infima, attainment of, 300-304 Infinite-dimensional effects problem, 216 Infinite-dimensional manifolds, 215, 356 Infinity, decay of nonlinear systems at, 21 Instability, dynamic, 15, 322-324 Integra curvatura, 341 Integrable almost complex structures, 147 Integrable systems, 4 Integrability condition, 9 Integral operators, 79-80 Integration, in elementary calculus, 65-67 Invariance of domain theorem, 269 Inverse, approximate, 137 lnverse function theorem, 113-1 15 for local conjugacy problems, 142-145 Irrationality conditions, 24 Ising model, 16 Isometric embedding problem, 145-146 Isoperimetric problems, 299, 324-336 in geometry and physics, 337-353 Peano’s theorem and, 122 symmetrization of functions in, 344 Isoperimetric variational problem, 18 steepest descent method for, 129-130 J Jacobi’s ellipsoids, 151 Jeffrey-Hamel flows, perturbed, 207-21 1

K Kahler manifolds, 313-314, 397 Kepler two-body problem, 10, 339 Klein-Gordan equation, 14 Knot configurations, in vortex rings, 353 Kondrachov compactness theorem, 42-43, 61 Korn-Lichtenstein theorem, 33 Korteweg-Devries equation, 61

L Laplace-Beltrami operator, 6, 343, 396-397 Laplacian, complex, 314 Lax-Milgram lemma, 34 Least potential energy, principle of, 182 Leray-Schauder degree, 246-257, 263, 267, 271,288, 290,296

414

INDEX

properties of, 250-25 1 infinite-dimensional, 356 Kahler, 313-314,397 LeviCivita, theory of, 340 Liapunov’s criterion, 153-154, 169, 337 Riemannian, 343-345,397 proof of, 157 Mappings Liapunov’s theorem, 291-292 compact homotopy and, 244 Linear elliptic differential operators, esticomplex analytic, 52 mates for, 45-46 cone-preserving, 28 1-28 3 continuous, 5 2 Linear Fredholm mapping, compact perturbacontinuous compact, 244 tions of, 256-262 Linear Fredholm operator, 161 contraction, 111-1 13 Linear homeomorphism, 116 degree of, 243-265 in Banach space, 113 between finite-dimensional spaces, 5 1-59 Linearization, 21-22 general smooth, 132 of general nonlinear operators, 217-231 gradient, see Gradient mappings Linearization problem Leray-Schauder degree of, 246-25 1 in bifurcation theory, 152 local analysis and, 111-149 proper, 102-106 in nonlinear functional analysis, 215 Linear operators singularities of, 125 compact, 35 with singular values, 225-231 Fredholm, 36 successive approximations in, 111-127 mapping properties of, 34-35 suspension of, 55 self-adjoint, 37 Maps, transition, 357 Linear stability theory, 152 Mathematical economics, 60 Lion’s lemma, 35 Mathematical physics Ljusternik-Schnirelmann category, 3 70-3 71 classical, 10- 14 Ljusternik-Schnirelmann critical point theory, contemporary, 14-16 353,366-374 nonlinear differential systems in, 9-16 Ljusternik-Schnirelmann invariants, in equisingular perturbation problems of, 204-21 1 variant mappings, 389 Mean field approximation, 14 Ljusternik-Schnirelmann multiplicity theorem, dynamic instability in, 322-324 368 Mean value theorem, in elementary calculus, Local analysis 70 continuous descent in, 128-1 29 Meromorphic functions, value distribution of, 8 Newton’s method in, 116-118 of single mapping, 111-148 Metric(s) successive approximations in, 111-1 27 conformally equivalent, 7 Local conjugacy problem, 109 Hermitian, 313-317, 397 inverse function theorems for, 142-145 with prescribed curvature properties, 7 Local mapping properties, in nonlinear opera- Minimal surfaces, 5 tor, 109ff Minimax principle, 38, 367-370, 375 Local sujectivity, 118-119 for quadratic functionals, 376 Lorentz metric, 15 Minimax theorem, 368 Lower semicontinuity, 5 Minimization problems attainment of infima in, 300-304 M in critical point theory, 299-312 from geometry and physics, 313-314 Maclaurin’s ellipsoids, 151 lower semicontinuity in, 300-301 Majorant method, 59 partial differential equations in, 313-314 analytic operators and, 133-137 quasilinear elliptic equations and, 306Majorant series, 133 312 Manifolds Minorants, monotone, 282-283, 292 complex, 394-395 Morse critical point theory, 58, 353-366, differentiable, 391-395 see also Critical point theory

INDEX

Morse inequalities, 361-363 Morse lemma, 156, 355 Morse theorem, 5 2 for geodesics between two points of compact Riemannian manifold, 388 Morse type numbers, 58, 63, 171, 358-361 small perturbations of, 171 Motion Euler equations of, 12 vortex, 12 Multilinear operator, polarization of, 96 Multiplicity preservation theorem, 376 applications of, 390

415

membrane approximation in, 206-207 stable equilibrium states in, 317-322 Nonlinear Fredholm mappings, regular and singular points and values in, 100 Nonlinear Fredholm operator index in, 99 rank theorem for, 147-148 Nonlinear free boundary value problem, 287 Nonlinear growth, versus dimension, 20 Nonlinear Hamiltonian system, 23 periodic solutions for, 337-340 “Nonlinearly perturbed” harmonic oscillators, 173 Nonlinear operators, 64-108, see also N Operators Fredholm, 99-101 Natural boundary conditions, extremals and, general, see General nonlinear operators 17 gradient mappings and, 93-99 “Natural constraint,” concept of, 331-332, higher eigenvectors for, 328 341 homotopy and, 266-283 Navier-Stokes equations, 13, 107, 183 local mapping properties and, 109ff Couette flow and, 186 mapping properties of, 266-283 Jeffrey-Hamel flows and, 207-208 specific, 76-84 secondary steady flows for, 183-188 Nonlinear problems, origin of, 3-18 steady-state solutions for, 241-243 Nonlinear systems Navier-Stokes operator, abstract, 107-108 difficulties encountered with, 18-24 Negative constant Gaussian curvature metncs, functional analysis of, 25-39 278 inequalities and estimates in, 39-47 Neumann problem, 80 nonuniqueness property of, 18-19 Newtonian mechanics of particles, 10-11 parametric dependence in, 19 Newton method, in local analysis, 116-1 18 singularities of, 19 Nonautonomous Hamiltonian system, 304 Nonlinear wave equations, stationary states Nonaxisymmetric equilibrium, 2 1 for, 383-386 Nonintegrable systems, 1 Nonquadratic functionals, 3 Nonlinear boundary value problems Nonstable homotopy groups, 261 conjugate harmonic functions and, 289 Nonuniqueness property, of nonlinear general nonlinear operators and, 283-295 systems, 18-19 Nonlinear differential systems Normal modes, problem of, 23 a priori bounds and, 23 0 asymptotic solutions in, 22-23 from calculus of variations, 16-18 Odd multiplicity eigenvalues, bifurcation and, decay of at infinity, 21 21 1 from mathematical physics, 9-16 Open mapping theorem, 34 nonintrinsic difficulties in, 21-24 Operator(s) resonance phenomena in, 23-24 analytic, 84-88 small divisor problems in, 22 compact, 88-93 Nonlinear effects problem, 109, 216 compact differential, 92 Nonlinear eigenvalue problems, 149, 287 completely continuous, 35 for gradient mappings, 325-331 composition, 76-77 Ljusternik-Schnirelmann theory and, 371-374 conepreserving, 281-283 for nonlinear operators, 272-277 convolution, 142 Nonlinear elasticity differentiable, 68-70

416

INDEX

differential, 77-84, 105-106 divergence form of, 77 Fr6chet differentiable, 67, 72 Gateaux differentiable, 68, 86 general nonlinear, see General nonlinear operators gradient, 94-99 Fredholm, see Fredholm operators integral, 79-80 linear, 34-35 multilinear, 70-72, 96 nonlinear, see Nonlinear operators projection, 38-39 quasilinear, 77 real analytic, 107 self-adjoint, 37 semilinear, 77 smoothing, 92, 141-142 Operator equations, multiple solutions of, 378-380 Oscillators, “nonlinear perturbed” harmonic, 173 P Parallelogram law, 26 Parameter dependence problem, 109 Parameter dependent perturbation phenomenon, 149-213 Partial derivatives, 69 Particles, Newtonian mechanics of, 10 Partition of unity, 391 Peano’s theorem, 122, 221 Periodic motions, continuation of, 291-293 Periodic water waves, 286-291 limiting form at crest of, 288 of mathematical physics, 204-21 1 Perturbation problems, 149-213, see also Compact perturbations Phase transitions, in solids, 16 Physics, mathematical, see Mathematical physics Plateau’s problem, 5, 319-322 Pointwise estimates, in Holder space, 40 Poisson equation, 27 Potential energy, least, 182 Potential theory problem, 27 Product rule, in elementary calculus, 69 Projection operators, 34, 38-39 Proper Fredholm mapping, 265 Proper Fredholm operator, 298 generalized degree for, 263 Proper mappings, 108

basic properties of, 103-105 coercive, 102 differential operators as, 105-106 local invertibility of, 104 Proper nonlinear Fredholm operators, linem ization of, 296

Q Quantum field theory, 14, 324 Quasilinear elliptic differential operators, 235236 Quasilinear elliptic equations Dirichlet problem for, 283-285 minimization problems associated with, 306312 Quasilinear elliptic partial differential equations, 297

R Rayleigh-Ritz approximations, for gradient operators, 239-241 Real analytic operators, 107 Reduction lemma, 226-227, 261 Regularization theory, 10, 340 Rellich’s lemma, 32 Resonance effects, 11, 23-24 Reynolds number, 107 Ricci tensor, 7, 15, 314 Riemannian manifolds, 145 geodesics between two points on, 386-387 with prescribed scalar curvature, 343-345 2N-dimensional, 397 Riemannian metric, 4, 396 Riemannian structures, of prescribed Gaussian curvature, 340 Riemannian surfaces, uniformization of, 6 Riesz representation theorem, 30-31, 289 Riesz-Tamarkin theorem, 32 S Saddle points, 17 in gradient mapping, 131 Saddle point solution, in isoperimetric problems, 324, 331 Sard’s theorem, 52, 125 Scalar curvature function, 7 Schauder fixed point theorem, 90,269-270 Schauder inversion, 81-82, 108 Schauder-type estimates, 46-47

417

INDEX

Schwarzchild metric, 15 Secondary stationary flow, 183-186 Self-adjoint operators, 37 spectrum of, 37-38 Semicontinuity, weak lower, 300-302 Semi-Fredholm operator, 36 Semilinear gradient operator equations, solvability of, 331-336 Simple multiplicity, bifurcation theorem for, 155-156 Singularity free solutions of nonlinear partial differential equations, 62 in nonlinear systems, 20 Singular perturbation problems, of classical mathematical physics, 204-21 1 “Singular” vortex ring, 348 Small-divisor problems, in nonlinear systems, 22 Smoothing operators, 141-142 Sobolev imbedding theorem, 35 1 Sobolev integral operator, 33, 36 Sobolev spaces, 28-30, 294, 350-351 Solids, phase transitions in, 16 Solvability, necessary and sufficient condition for, 336 Spheres, homotopy groups of, 55 Stability problem, 109 Stable homotopy, 256-262 Stable homotopy class, 259, 279 Stationary states, for nonlinear wave equations, 383-386 Steepest descent method, 59 for gradient mappings, 127-1 33 sharpening of, 354-355 Steiner symmetrization, 345 Stokes periodic “limit” waves, 288 Stokes stream function, 1 3 Strange particles, 14-15 Stream function, 1 3 Stress-strain laws, 11 Superconductivity, vortex rings and, 353 Superfluidity, vortex rings and, 353 Superposition principle, 14, 35 3 Surjectivity properties, for nonlinear operators, 266-267 Svarc’s theorem, 258 Symmetrization, 344-345 Symmetry-breaking phenomena, 5

T Taylor’s theorem, 74-76 Taylor vortices, 186 Thin elastic shells, buckling phenomena in, 180 Three-body problem, 10 periodic motions near equilibrium points in, 173-177 Truncation Fourier series, 142 Turbulence, 14 Two-body problem, 10, 339

U Uniform boundedness theorem, 34 Uniformization problem, 224 Unity, partition of, 391 Univalence, for general nonlinear operators, 268-269 Utility function, 60

V Variational principles, in critical point theory, 299 Variations, calculus of, see Calculus of variations Von K k m i n equations, 11, 97, 318 Von Khmin quasilinear partial differential equations, 177 Vortex motions, 12 Vortex ring distributions, 1 3 Vortex rings, 12 nonaxisymmetric, 35 3 of small cross section, 353 steady, 348-350 Vorticity, defined, 12

W Water waves, periodic, 286-291 Weak convergence, 31 Weak solutions, for nonlinear problems, 4851 Weighted norms, 61 Z

Zygmund’s theorem, 289

Pure and Applied Mathematics A Series of Monographs and Textbooks Editors

Samuel Ellenberg and Hyman Bass Columbia University, New York

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